Depends on the mathematicians (Score 2, Interesting)

That completely depends on the mathematicians, and the kind of mathematics they do. For proofs that rely only on calculations, you do not need even to understand the low dimension case, just do the computations right.

But proofs with computations are rarely elegant. Some mathematicians prefer a more geometric approach, and for that, they need to see, un to a certain level, the objects in higher dimensions.

Furthermore, the 2D or 3D spaces we have direct access to are really limited. There are lots of phenomenas that only happen starting with dimension 4 or 5. For example, think of this 2D property: "two lines perpendicular to a common third line are parallel"; if you try to take it as is in higher dimensions, you get something false; fortunately, you can think in 3D and see that it is false. There are similar examples in higher dimensions. Curvature, for example: curvature of 2D surfaces in 3D spaces is misleadingly simple, compared to curvature of higher dimensional spaces.

Sometimes, there just is not space enough to build the objects you need in 3D space. For example, if you want to study circles drawn on a sphere, the object you need to make the properties apparent is a 3D hyperboloid in a 4D space. If you settle for a 2D hyperboloid in a 3D space, you end up studying pairs of points on a circle, which is rather boring.