Intuitive explanation behind Goemans-Williamson randomized rounding A very simple randomized cut algorithm achieves $1/2$ of the optimal value: just choose each vertex to be in the cut with probability $1/2$, independently. Goemans-Williamson does something more sophisticated: first, they do a semi-definite relaxation to obtain $n$ unit vectors $X_i$ (rather than the $n$ bits $x_i$ that the original integer program MAXCUT is supposed to solve for), and then each vector  $X_i$ is converted to a random bit by seeing what side of a random hyperplane
$g\sim N(0,I_n)$
 it falls on.

Intuitively, the new random variables $\mathrm{sign}(X_i\cdot g)$ are no longer independent -- unlike the naive randomized cut, where each vertex was chosen independently. I can see why creating dependencies might be beneficial: if a given vertex $i$ is chosen to be in the cut and has a large degree, then perhaps we might not want its neighbors to be in the cut. Is there an intuitive explanation behind the kind of dependencies that Goemans-Williamson rounding is creating between the vertices and why these are beneficial for obtaining larger cuts? [I understand the formal proof, which is relatively simple and very slick. Looking for intuition!]
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.