One thing I've learned recently (moving into symplectic geometry from topology) is that people often underestimate the value of regarding manifolds locally as graphs of functions, or submanifolds of Euclidean space. The IFT helps with this, and provides actual applications, but here's possible motivation.

--My recent favorite example for intuition: How does a surface of negative curvature behave? Like the graph of a harmonic function on $\mathbb R^2$ (where the Hessian is not 0). That is to say, if you have trouble imagining a surface in $\mathbb R^3$ which has no hills and valleys but only saddles, try graphing a function like $xy$ or $e^x\cos y$.

--Related to this: use IFT to study the curvature of a surface in $\mathbb R^3$ (or $n$-manifold in $\mathbb R^{n+1}$) nearby a strict maximum in any one coordinate. I'm not sure what the nicest precise statement should be here to cover all non-generic cases, but it's fun route to differential geometry if someone hasn't had it.

--Less directly related to the IFT: What does a Morse critical point "look like''? We like to say that the Morse function can be given in some local coordinates by $x_1^2+\ldots +x_k^2-x_{k+1}^2-\ldots x_n^2$, but when we visualize this, we usually see the manifold (OK, surface) itself as the graph of that function over a piece of $\mathbb R^n$. Thus making the Morse function "height" as usual.

local immersion theorem. (The implicit function theorem [slightly extended, compared to some treatments] is then thelocal submersion theorem, of course.) This terminology also appears in Abraham, Marsden, and Ratiu (1988), for example. $\endgroup$ – Alex Shpilkin May 8 '17 at 23:35