The intuitive way to think about it is that with very few dimensions you have very few degrees of freedom, so it's easy to prove things possible or impossible.

The intuitive way to think about it is that with very few dimensions you have very few degrees of freedom, so it's easy to prove things possible or impossible. With lots of dimensions, you have enough wiggle room to prove most things possible. Somewhere in between, you have enough complexity to not trivialize the problems but not enough wiggle room to be able to easily circumvent the issue. Often in practice, that boundary is around 3-4 dimensions. See the poincaré conjecture, various sphere packing shenanigans, graph embeddings, ....