In simple words, he is saying that the other man snores in a particular way (different from the way other people snore/sound) -- maybe with certain sounds, presumably. To say that there's meaning in the snores suggests that he can (in his mind, anyway) detect a sort of language or communication.
Explanation:
'Distinctly' indicates in a way that is readily and clearly distinguishable by the senses -- something that is very noticeable or apparent.
