Why polyester clothes dry quickly?

Because it's made out of gasoline (well, oil).

Actually, compared to other common clothing materials, polyester doesn't burn well. That is, it tends to burn very slowly, and prefers to melt when exposed to flame. This is usually the result of the chemical structure of the particular polyester molecule, as most polyester molecules are "tightly structured"; that is, they have multiple bonds between atoms, making it much harder for them to be split and oxidize. Rather than split the polyster molecule (or, be able to add an OH radical), the addition of heat tends to force a realignment of multiple molecules in the polyester fabric - thus, they melt first, rather than ignite.

Overall, this is tendency to melt rather than burn is reflected in the much higher heat capacity of polyester relative to other common fabrics (cotton, rayon, etc.).