Intuitive explanation behind Goemans-Williamson randomized rounding
Great intuitive explanations above by Sasho Nikolov and Neal Young. Here's another one Eden Chlamtáč emailed in:

Well, the intuition is that the SDP solution assigns greater weight in
the objective function to edges whose endpoints that are placed
farther apart in the vector solution, so it makes sense to cluster the
set of vertices geometrically into two contiguous sets. A random
hyperplane is the simplest way of doing that, and it happens to work
well.