Grigory Perelman didn't want the money or the fame—he loved math for math's sake.
The provenance of Perelman's unclaimed Fields medal was a question posed in 1904 by Henri Poincare, the celebrated French mathematician. The famous question, known as Poincare's Conjecture, can be worded succinctly in mathematical parlance like so: "every simply-connected closed three-manifold is homeomorphic to the three-sphere." (Here's the official description from the Clay Institute—more on them later.)
The "three-sphere" in question is the skin of a four-dimensional sphere, just as a two-sphere is the curved surface of a three-dimensional sphere. Since I can't for the life of me image what a four-dimensional sphere looks like —though the language of mathematics, useful thing that it is, allows folks to consider and play with these non-existent objects—I find it easier to consider the problem in dimensions I can visualize.
People who care about problems like these are called topologists; I think of them as the type of mathematicians who never tired of the joys of playdoh. Given a doughnut of playdoh, they can mold it into a coffee mug or a teacup without having to tear it or stick pieces together together. Similarly, they can mold a sphere into a football, bend it into a bowl, or smash it into a plate. But turning the plate into a doughnut requires tearing a hole in the middle, and doing the opposite requires sticking pieces of the doughnut together to close that hole.
What I'm trying to get at with all this nostalgic talk of playdoh is an intuitive understanding what "homeomorphic" means. The coffee mug and doughnut are essentially the same form, so they are homeomorphic, as are the bowl and the sphere. But the doughnut can never be molded into the sphere, so these are two fundamentally different forms. Next time you come across some playdoh (I keep a little pot next to my computer in case I'm suddenly inspired to discover new topological truths) try it yourself.
What about "simply connected?" Imagine looping a string around the middle of a sphere. What happens when you pull one end of the string, tightening the loop? The loop will slip away from the middle, enclosing a smaller and smaller circumference, until it becomes a point on the surface. The same happens when you loop the string around a football. But if you loop the string through the ring of a doughnut, there's no way to shrink it down to a point without cutting it and tying it back together. So if a loop on a surface can shrink down to a single point without breaking, the shape is "simply connected."
You can smoothly shrink the rubber band around the sphere to a point, but you can't shrink the rubber band on the doughnut without breaking it.
In the dimensions we can visualize, Poincare's conjecture says that any surface on which you can shrink a loop down to a point—a football, bowl, or plate—must really be equivalent to a sphere. What Perelman did was rigorously prove that this was true if you take the whole problem to a higher dimension, so that the magic shape is the three-dimensional skin of a four-dimensional sphere.
On November 11, 2002, Perelman posted his 39-page proof on the preprint arxiv, without a care for publication in a peer-review journal or possible plagiarism. You can read "The entropy formula for the Ricci flow and its geometric applications" for yourself; Ricci flow is the mechanism Perelman used to smooth out bumps in flaw-ridden three-dimensional surfaces, revealing them to be equivalent to the smooth, curved three-sphere. His proof won him the Fields Medal (many think of it as math's Nobel prize) in 2006. The Clay Institute, who named Poincare's Conjecture as one of their seven "millennium problems", have worked on turning an explication of Perelman's proof into a published work so he can be eligible for the million dollar prize. (Jury's still out on whether Perelman will, if offered, take it.) Meanwhile, mathematics professors from all over the world aggressively tried to recruit him to their departments.
Perelman refused all offers. In the Wall Street Journal article, Masha Gessen, who has written a book about Perelman, tries to psychoanalyze both Perelman's successful proof and his refusal to accept the Fields Medal as artifacts of his upbringing in Russia's mathematical counterculture. This was the world outside of institutionalized Soviet math, the stables of Russian mathematicians preened, nurtured, and controlled by the state.
Whether by religion, ethnicity, or political leanings, some mathematicians could never join this institution, but that didn't stop them from doing great work. Outside of state-run research towns, mathematics was pursued as a hobby or an art more akin to poetry than engineering, Gessen writes. It was a solo enterprise, independent of academia, teaching, publishing, or professional ambitions. Although Perelman spent time in the US as a post-doc in the early 90s, those mundane aspects of American math drove him back to Russia. There, in the relative solitude—he did have colleagues and his proof rests heavily on previous work—familiar to Russian mathematicians on the outside, he worked on his proof. And he seems content to stay on the outside.