Sarte's reasoning is pretty simple: he doesn't want to become a component of an institution. He wants to be able to write as Jean-Paul Sartre, not Jean-Paul Sartre the Nobel Prize winner.
That thinking is flawed though. A prize doesn't define you by accepting it. When I am given a birthday cake, I don't then become "heuving, the receiver of birthday cakes".
The only reason that really makes sense to decline a prize is if you are heavily opposed to what the institution represents. However, he didn't strike me as anti-intellectual.