He had probably previously encountered Cantor's diagonalization technique for proving the uncountability of the reals. He just saw how to apply the same technique to the question of describability.

It's important to note that the "diagonal argument" used to show countability of rationals or describable numbers is completely different from the one used to show the uncountability of the reals. In fact, the two counting arguments used to show countability of rationals and describable numbers are themselves quite different (I would say the latter isn't even a diagonal argument at all; it just goes through a countable union of finite sets one by one).