How should you explain parallel transport to undergraduates? The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.
This is in the vein of many other questions on mathoverflow:

*

*What is the Levi-Civita connection trying to describe?


*What is torsion in differential geometry intuitively?


*Rolling without slipping interpretation of torsion
But the focus is different.
Let me summarize my understanding given answers in the questions above:
(Very Rough) Summary of Answers in Previous Questions

*

*There exists an interpretation in terms of $G$-structures, as in the chosen answer (by Chris Schommer-Pries) to What is torsion in differential geometry intuitively?.

*There exists an interpretation as having some universal property, as described in the top answer (by Robert Bryant) to What is the Levi-Civita connection trying to describe?.

*There exists a deceiving but appealing interpretation by parallelograms whose sides don't really lie in the same space, as in the answer by Gabe K to What is the Levi-Civita connection trying to describe?. (Gabe K did a heroic effort to make sense of the nonsensical diagram, and I thank him dearly.)

*There exists an interpretation regarding rolling the shape on a surface (Rolling without slipping interpretation of torsion).

But ultimately, none of that is something that I can intuitively sell to an undergraduate, and by undergraduate I really mean my heart. In the bottom of my heart, I need a better explanation, one that starts with desirable properties, and then proceeds through existence and uniqueness.
Outline of the Type of Intuition I Desire
I want to start with some desirable behaviors, which I allow to be external (i.e, to reference a given embedding of the Riemannian manifold into $\mathbb{R}^n$), and then say that the only notion of parallel transport that satisfies these conditions must be the Levi–Civita connection. (Any reasonable notion of parallel transport will respect the metric, so I'm really thinking of the torsion-free condition.)
A base case of a desirable condition is that for the Riemannian manifold $\mathbb{R}^n$, parallel transport is the trivial thing. (If one identifies the tangent bundle with $\mathbb{R}^n\times\mathbb{R}^n$ then for any path $\gamma$ the parallel transport of the tangent vector $(\gamma(0),v)$ at $\gamma(0)$ to $\gamma(1)$ via $\gamma$ is the tangent vector $(\gamma(1),v)$ at $\gamma(1)$.)
Next, we would like some way to generalize to a general Riemannian manifold. Let $(M,g)$ be a Riemannian manifold, and let $p\in M$ be a point. Then by the implicit function theorem we can have a chart $f:V\rightarrow U\subset \mathbb{R}^d$ where $0\in V\subset \mathbb{R}^n$, and $p\in U\subset M$, such that $f(0)=p$ and such that $f$ is the identity on the first $n$ coordinates.
My next thought is to look at the most intuitive case of torsion-freeness, which is the case of commuting fields $X$ and $Y$. By change of coordinates, we can assume WLOG that on $V$ the vector fields $X$ and $Y$ are defined via the constant functions $X(v)=e_1$ and $Y(v)=e_2$. One can then express $X$ on $Y$ on $M$ via the derivative of $f$.
But I'm missing multiple components to proceed.
So let me ask this in terms of several more explicit questions.
Questions

*

*If a connection satisfies that $\nabla_XY=\nabla_YX$ for any commuting set of vector fields $X$ and $Y$, then is it torsion-free? (In other words, if you're torsion-free on commuting vector fields, are you torsion-free for all vector fields?)

*What intuitive desirable condition (that is allowed to use a given embedding of $M$ into some $\mathbb{R}^d$), combined with, or perhaps generalizing the desired behavior of parallel transport on Euclidean space, would uniquely determine it as satisfying $\nabla_XY=\nabla_YX$ for commuting vector fields? (Perhaps something about geodesics? Or volumes? I don't really know what's the missing component here.)

*I feel like Ben McKay's answer to What is the Levi-Civita connection trying to describe? is coming close to what I want, but I did not get to the bottom of it. It appeared at first that he was saying that the Levi–Civita parallel transport is simply parallel transporting in the ambient space, and then projecting to the tangent plane. But in retrospect, my interpretation is clearly wrong. (Imagine for example an upward pointing vector on the equator of a sphere, being parallel transported to the top. If you parallel transport in $\mathbb{R}^3$ you'll get a vector pointing up, which projected to the tangent space will be the $0$ vector.)

*A little more vaguely, in case you have an entirely different notion in mind, how would you explain parallel transport to the undergraduate in your heart?

 A: I prefer to introduce the Levi-Civita connection before the concept of parallel transport. On the surface, parallel transport seems like it should be more intuitive and easier to visualize geometrically. On the other hand, as you yourself indicate, it's not so obvious.
Although I have a preference for introducing manifolds abstractly and not as a submanifold of Euclidean space, I prefer to introduce Riemannian geometry starting with the induced geometry of a hypersurface in Euclidean space. The Levi-Civita connection arises naturally when you study the directional derivative of a vector field. It arises from two important observations: First, the directional derivative splits into tangential and normal components. Second, the normal component depends only on the (second fundamental form) surface and not on the curve or the derivative of the vector field. So it is natural to define the directional derivative of the vector field as only the tangential component.
Now you study this directional derivative. It has (at least) two nice properties: 1) As Willie mentioned, if you define the Hessian of a function using it, the Hessian is symmetric. 2) It is compatible with the metric and torsion-free. More importantly, it is the unique connection satisfying these two properties. This means that any property of the connection is a property of the metric.
This now makes the concept of the Levi-Civita connection for an abstract Riemannian manifold a compelling one. Parallel transport is now a natural development from that.
A: I like the bike wheel interpretation introduced by Mark Levi (A “bicycle wheel” proof of the Gauss–Bonnet theorem Exposition. Math. 12.2 (1994), 145–164).
It takes no time in class and helps to build right intuition.
A: This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe.  Here is a more explicit, extrinsic description that may help:
Suppose that $M^m\subset\mathbb{E}^n$ is an isometrically embedded submanifold of Euclidean $n$-space.  Let $\gamma:(a,b)\to M^m$ be a smooth curve in $M$ and let $v:(a,b)\to\mathbb{E}^n$ be a curve of vectors along $\gamma$, i.e., $v(t)$ lies in the tangent space $T_{\gamma(t)}M$ for all $t\in (a,b)$.  Say that $v$ is parallel (along $\gamma$) if $v':(a,b)\to\mathbb{E}^n$ is normal to $TM$ along $\gamma$, i.e., $v'(t)\perp T_{\gamma(t)}M$ for all $t\in(a,b)$.  In other words, the velocity of $v$ is always perpendicular to the tangent vectors to $M$ at the point of tangency.
Then the (easily proved) proposition is that this notion of a tangent vector field along a curve  being parallel along $\gamma$ does not depend on the choice of the isometric embedding, i.e., it is intrinsic to the metric induced on $M$ by its embedding.  More generally, if $v:(a,b)\to\mathbb{E}^n$ is tangent along $\gamma$, then letting $D_\gamma v(t)$ be the orthogonal projection of $v'(t)$ onto $T_{\gamma(t)}M$ yields another curve $D_\gamma v:(a,b)\to\mathbb{E}^n$ that is tangent along $\gamma$, and this operation (actually a derivation) on tangent fields along $\gamma$ depends only on the induced metric on $M$.  Since it is independent of the choice of isometric embedding, it is the 'covariant part' of the ambient derivative, i.e., the 'covariant derivative'.
For example, it follows from the definition that if $v$ is a parallel tangent vector field along $\gamma$, then the length of $v$ is constant.  Then the existence and uniqueness of 'parallel transport' follow by elementary ODE arguments. The Leibnitz rule for the 'covariant derivative' and other properties are easily derived from the definition as well.
Once you know that $\nabla_{\gamma'}v$ for a curve of tangent vectors depends only on the metric, it's natural to want to find a formula for it that uses only on the metric and not the (superfluous) isometric embedding.  That is what leads to the usual characterizations.
A: I agree with Ben McKay and Robert Bryant that the best way to introduce parallel transport to students, or to provide some motivation and intuition for it, is via an extrinsic approach, i.e., by the example of a tangent vector field to a surface whose derivative along a curve is orthogonal to the surface. Then I explicitly compute for them the parallel transport along a meridian of a sphere in an elementary way which I attach below.
After I do this computation I tell the students that this explains the precession of the swing plane of a pendulum, as observed by the French Physicist Leon Foucault in $1851$ (which predates Levi-Civita; see the historical note below), and how they can use the answer to figure out the latitude of their location on Earth. There is a  nice Wikipedia page on Foucault's Pendulum which I also tell the students to check out.
Finally, I will tell them that the computation can be carried out much more quickly via the well-known trick of constructing a cone which is tangent to the sphere along the meridian and unrolling the cone into the plane. Since the cone and the sphere are tangent along the curve, a tangent vector field along the curve is parallel with respect to one surface if  it is parallel with respect to the other one. Furthermore since, as Robert mentioned, parallel transport is intrinsic,  it is not effected by unrolling (or isometric immersion) of the cone into the plane, where parallel transport is trivial, and the total rotation of the vector field with respect to the meridian can be computed immediately (it will be equal to the total angle of the cone at its apex).
Here is the  explicit computation for the parallel transport of a vector along a meridian of the unit sphere $\mathbf{S}^2$. This is an excerpt from my lecture notes.  Let
$$X(\theta,\phi):=\big(\cos(\theta)\sin(\phi), \sin(\theta)\sin(\phi), \cos(\phi)\big)$$
be the standard parametrization for $\mathbf{S}^2-\{(0,0,\pm 1)\}$. Suppose that we want to parallel transport a given unit vector $V_0\in T_{X(\theta_0,\phi_0)}\mathbf{S}^2$ along the meridian $X(\theta,\phi_0)$. So we need to find a mapping $V\colon[0,2\pi]\to\mathbf{R}^3$ such that $V(0)=V_0$, $V(\theta)\in T_{X(\theta,\phi_0)}\mathbf{S}^2$, and $V'(\theta)\perp T_{X(\theta,\phi_0)}\mathbf{S}^2$. The latter condition is equivalent to the requirement that
$$
V'(\theta)=\lambda(\theta) X(\theta,\phi_0), \quad(*)
$$
for some scalar function $\lambda$,
since the normal to $\mathbf{S}^2$ at the point $X(\theta,\phi)$ is just $X(\theta,\phi)$ itself. To solve this equation let
$$
E_1(\theta):=\frac{\partial X/\partial\theta(\theta,\phi_0)}{\|\partial X/\partial\theta(\theta,\phi_0)\|}=\big(-\sin(\theta),\cos(\theta),0\big),
$$
and
$$
E_2(\theta):=\frac{\partial X/\partial\phi(\theta,\phi_0)}{\|\partial X/\partial\phi(\theta,\phi_0)\|}=\big(\cos(\theta)\cos(\phi_0),\sin(\theta)\cos(\phi_0),-\sin(\phi_0)\big).
$$
Note that $\{E_1(\theta),E_2(\theta)\}$ forms a basis for $T_{X(\theta_0,\phi_0)}\mathbf{S}^2$. Thus equation $(\ast)$ above is equivalent to
$$
\langle V'(\theta), E_1(\theta)\rangle =0\quad\text{and}\quad \langle V'(\theta), E_2(\theta)\rangle =0. \quad (\ast\ast)
$$
It remains to solve this system of differential equations. Since $V(\theta)\perp V'(\theta)$, $V(\theta)$ has unit length. So  we may write
$$
V(\theta)=\cos(\alpha(\theta))E_1(\theta)+\sin(\alpha(\theta))E_2(\theta),
$$
for some angle function $\alpha$.
Differentiation yields that
$$
V'=E_1'\cos(\alpha)-\sin(\alpha)\alpha'E_1+\sin(\alpha)E_2'+\cos(\alpha)\alpha' E_2.
$$
Furthermore, it is easy to compute that
$$
E_1'=-\cos(\phi_0) E_2-\sin(\phi_0)E_3\quad\text{and}\quad E_2' =\cos(\phi_0) E_1,
$$
where $E_3(\theta):=X(\theta,\phi_0)$. Thus we obtain:
$$
V'=\sin(\alpha)(\cos(\phi_0)-\alpha')E_1+\cos(\alpha)(\alpha'-\cos(\phi_0))E_2 +(some\; terms) E_3.
$$
So for ($\ast\ast$) to be satisfied, we must have $\alpha'=\cos(\phi_0)$ or
$$
\alpha(\theta)=\cos(\phi_0)\theta+\alpha(0),
$$
which in turn determines $V$. Note in particular that the total rotation of $V$ with respect to the meridian $X(\theta,\phi_0)$ is given by
$$
\alpha(2\pi)-\alpha(0)=\int_0^{2\pi}\alpha'd\theta=2\pi\cos(\phi_0).
$$
Thus
$$
\phi_0=\cos^{-1}\left(\frac{\alpha(2\pi)-\alpha(0)}{2\pi}\right).
$$
The last equation gives the relation between the total precession of the swing plane of a pendulum during a $24$ hour period, and the longitude of the location of that pendulum on Earth, as observed by  Foucault.
Historical Note: The precession of pendulum was mentioned by the Dutch geometer Jan Arnoldus Schouten in a letter to Levi-Civita in 1918, where he wrote "a remarkable physical application is the Foucault pendulum". This paper appears to give a nice account of the historical development of parallel transport through the correspondence between Schouten and Levi-Civita.
A: If intuition is what you want then I think the essence of the matter is best expressed using locally flat coordinates at each point as follows. At any point $p$ in a Riemannian (or pseudo-Riemannian, for those of us who use this in the context of general relativity) manifold, one can always choose "locally flat coordinates" (lfc) such that the metric components gave vanishing first partial derivatives at $p$. The (Levi-Civita) covariant derivative of a vector (or tensor) at $p$ is simply the partial derivative with respect to any lfc. A vector is parallel transported along a curve if at each point in lfc its components have vanishing derivative with respect to the curve parameter. This is of course not a practical definition, since the lfc are different at each point. For practicality, one can use the connection components in any coordinate system. Note however that if the curve is a geodesic, then there exist coordinates that cover a neighborhood of the curve and are locally flat at every point on the curve. In that case, parallel transport just keeps all the components of the vector constant in such a coordinate system. An example is transport around the equator of the 2-sphere using standard spherical polar coordinates.
