"Rolling without slipping" is a powerful idea, but the phrase doesn't necessarily lead one to the intended mental model. In particular, torsion is something that is at issue only for manifolds of dimension 3 or higher. Perhaps you can imagine taking a 3-manifold, and rolling it along a hyperplane in 4-space --- but the metaphor becomes strained, partly because most Riemannian 3-manifolds cannot be smoothly isometrically embedded in $\mathbb E^4$.
Another way to think of it is this: suppose you have a smooth parametrized curve say in a Riemannian 3-manifold, $\alpha: [0,T] \rightarrow M^3$. Then the claim is that there exists exists a matching curve $\beta: [0,T] \rightarrow \mathbb E^3$ together with a map $\phi$ from a neighborhood of the image of $\beta$ to a neighborhood of the image of $\alpha$ that takes the Euclidean metric to the metric of $M^3$ up to first order along the curve. Furthermore, $\beta$ is uniquely determined up to an isometry of $\mathbb E^3$. Basically, $\beta$ is what you get if you "roll" $M^3$ along $\mathbb E^3$ along $\gamma$ in a way that best maintains contact between the two spaces: that is, "without slipping".
To see that the curve $\beta$ (assuming it exists) is uniquely determined by $\gamma$, \alpha$, we can imagine trying to send a neighborhood of $\beta$ to a neighborhood of another curve $\gamma: [0,T] \rightarrow \mathbb E^3$ in a way that maintains first order contact of the metric. If you look at curves parallel to $\beta$ in $\E^3$, \mathbb E^3$, the first derivative of their arc length is negative in the direction that $\beta$ is curving. The logarithmic derivative of arc length, for curves displaced along a normal vector field that remains as parallel to itself as possible, is the magnitude of the curvature.
If you try to twist the normal coordinate system, this corresponds to the concept of torsion. It's easiest to visualize along a straight line: if you twist a neighborhood of a line in space, you distort the metric on each concentric cylinder, by changing the angles between cross-section circles and generating lines. I.e., threads that wind around a hose at angles $\pm \pi/4$ are effective at preventing twisting (= torsion). The same principle holds for any curve in space: the first order behavior of the metric in a neighborhood of the curve locks in the Frenet characterization of the curve (well, the curvature and torsion as a function of arc length, but these are different from but related to the curvature and torsion of a connection).
Why does the matching curve exist? You can check derviatives etc. but better to just imagine it. Basically, you could reparametrize $\alpha$ by arc length, then map project a neighborhood of $\alpha$ back to $\alpha$ by sending each point to the closest point of $\alpha$ \alpha$, and parametrize the lines of projection by their arc length. On each concentric tube, there's a unique unit vector field orthogonal to the preimages of projection to $\alpha$. Scale this vector field so that it commutes with projection, to get a full set of cylindrical coordinates for a neighborhood of $\alpha$. The only first-order invariant for the metric that is free is the first derivative of scaling function. Using that, you can match the first derivative by using the curvature and torsion of a curve in space.
This process defines the affine connection on the tangent bundle. The Levi-Civita connection is the linear part of the affine connection, which is automatically by definition torsion free. The non-torsion-free connections are ones that impart twists on little neighborhoods of curves. This is usually expressed by translating it into a formula about covariant derivatives of two vector fields not being as commutative as it should be.
This really calls for pictures. Any volunteers?