Here are some of the crutches I've relied on. (Admittedly, my crutches are probably much more useful for theoretical computer science, combinatorics, and probability than they are for geometry, topology, or physics. On a related note, I personally have a much easier time thinking about $R^n$ than about, say, $R^4$ or $R^5$!)
If you're trying to visualize some 4D phenomenon P, first think of a related 3D phenomenon P', and then imagine yourself as a 2D being who's trying to visualize P'. The advantage is that, unlike with the 4D vs. 3D case, you yourself can easily switch between the 3D and 2D perspectives, and can therefore get a sense of exactly what information is being lost when you drop a dimension. (You could call this the "Flatland trick," after the most famous literary work to rely on it.)
As someone else mentioned, discretize! Instead of thinking about $R^n$, think about the Boolean hypercube $\lbrace 0,1 \rbrace ^n$, which is finite and usually easier to get intuition about. (When working on problems, I often find myself drawing $\lbrace 0,1 \rbrace ^4$ on a sheet of paper by drawing two copies of $\lbrace 0,1 \rbrace ^3$ and then connecting the corresponding vertices.)
Instead of thinking about a subset $S \subseteq R^n$, think about its characteristic function $f : R^n \rightarrow \lbrace 0,1 \rbrace$. I don't know why that trivial perspective switch makes such a big difference, but it does ... maybe because it shifts your attention to the process of computing $f$, and makes you forget about the hopeless task of visualizing S!
One of the central facts about $R^n$ is that, while it has "room" for only $n$ orthogonal vectors, it has room for $\exp(n)$ almost-orthogonal vectors. Internalize that one fact, and so many other properties of $R^n$ (for example, that the $n$-sphere resembles a "ball with spikes sticking out," as someone mentioned before) will suddenly seem non-mysterious. In turn, one way to internalize the fact that $R^n$ has so many almost-orthogonal vectors is to internalize Shannon's theorem that there exist good error-correcting codes.
To get a feel for some high-dimensional object, ask questions about the behavior of a process that takes place on that object. For example: if I drop a ball here, which local minimum will it settle into? How long does this random walk on $\lbrace 0,1 \rbrace ^n$ take to mix?
