I'd like to highlight a point of dissonance between the title ("How mathematicians think...") and the actual request ("anything that makes it easier to see, for example, the linking of spheres") -- emphasis mine.
I find the identification between visualization and intuition revealing. As a rule, mathematicians must be able to reason about things they cannot even begin to visualize -- non-measurable sets, infinities so large they need special names, infinite linear combinations of orthogonal functions.
That's not to devalue attempts at visualization. They're useful for developing intuition. But the original joke works because the mathematician is perfectly happy reasoning in hyperspace even though he cannot see it. The fourth dimension is not particularly harder to describe than the nth.