Well, for starters he wasn't born deaf. He went gradually deaf as an adult, and well after he had already attained success as a composer. So, as a result, it's not like he didn't already have a great deal of musical knowledge and experience, having written a great deal of music for many instruments and voice. He would have a wealth of music information to draw on.

Also, as a musician, I can sit down with a piece of written music and just read it and know how it goes. I don't have completely perfect pitch, and am not the best pianist, but I can rehearse a piece from start to finish while just reading it and not singing a note. I don't need to hear it, because I've played enough music and sang enough that I can sing it in my head. And I'm not even that great at it.

Beethoven was an absolute genius. He could write entire symphonies and hear them in his head. So much so that Beethoven conducted the premiere of his 9th symphony, despite being completely deaf by that time.

While we're speaking of deafness and music, I should also mention that even being born deaf isn't necessarily a complete impediment to being a musician. Dame Evelyn Glennie is one of the world's foremost percussionists and has been profoundly deaf since she was a child. She learned to play by feel, and performs barefoot in order to feel the vibrations from the instrument. She also performs regularly with orchestras and other ensembles and doesn't require any hearing to stay in time with other musicians.