This isn't a question about how to visualize higher dimensions, or how intuit them, or how unintuitive they are.
Rather, it's a hypothetical question about the kinds of questions that might be easier to answer (not necessarily prove, but to suggest) if we could visualize $n$ dimensions as easily as we could 2 or 3. As an example, it's not hard to imagine we would probably have a better idea about kissing numbers if we could picture higher dimensions as easily as the euclidian plane.
What other kinds of (unsolved) problems might lend themselves to analysis more easily if we could intuit $n$ dimensions?