Intuition for Exotic $\mathbb R^4$'s

The intuition is basically that $4$ dimensions is large enough that $4$-manifolds exhibit great variety (one example: there are $4$ manifolds with arbitrary finitely-generated fundamental groups), but small enough that the high-dimensional-manifold surgery theory apparatus doesn't apply to smooth manifolds, only topological manifolds.

The key idea for the latter is the Whitney trick. This trick allows one to isotop apart embedded submanifolds of complementary dimension which algebraically have intersection number zero under certain conditions. The rough idea is to cancel intersection points of opposite signs in pairs by finding a Whitney disk, a disk with half its boundary on one submanifold and the other half on the other, meeting at the two points. Once you have one, if the Whitney disk and each submanifold has codimension greater than two, the whitney disk generically is embedded and doesn't meet either of them on its interior. Then you simply push one of the submanifolds across the disk. (There are other cases which work, but this is the simplest to explain.)

In dimension four this trick doesn't work, because the Whitney disk has codimension two, so it may have self intersections which can't be wiggled away, and may also intersect each other submanifold similarly, if they're each dimension two.

Bizarrely, one can make this trick work topologically (but not smoothly), roughly by iterating the construction infinitely many times. See: Casson handle and Freedman's theorem on these.

(Then Donaldson came along with an amazing way to tell smooth $4$-manifolds apart, but I'm not sure this belongs to the intuition regarding this.)