Most people who read music read the notes. They see the pitches and match them to their instrument. People with perfect pitch can see the note and immediately hear it in their head. (They are a strange, radical race. I happen to have a faction of friends who want to find all those with perfect pitch and breed them into a super-race, like Uruk-hai, but less tone-deaf. Speaking of Uruk-hai, apparently in between takes during the long night shoots all the extras in costume at Helm's Deep would break into song. So perhaps Uruk-hai are not as tone-deaf as first supposed. Has Tolkien given us any hard evidence to the contrary?) I've even heard of one perfect pitcher (no, not talking about Sandy Koufax) who actually passed out when Sara Brightman was singing at the Opening Ceremonies in Beijing because she was so off key. The point being, most musicians see, hear and learn music based off of the printed source.
There are also people who learn by ear. Call it the "suzuki" method. I happen to think using this as a basic teaching technique is bollocks, but it is a helpful method to enhance music memorization. (I can't knock it too much - the first thing I do when I need to learn a new song is find it on Rhapsody or YouTube.) I have friends in a choir who lean all their music by ear, and are given CD's of songs and told to hear and memorize the parts. Musicianship-wise, this isn't very quality, but it gets the job done and they do sound great at concerts.
They way I hear it, I believe, is in sections of "should's" and "shouldn'ts" (Shoulds for how this sections "should" sound, "shouldn'ts" for what wouldn't make sense in the section.) After hearing a song about three times, the song partitions itself into section of shoulds and shouldn'ts, branches of the song that look/sound one way or another and piece together to form a song as a whole, with modulation, variation and so forth. I see/hear these sections with a quasi-photographic memory. (It wants to be photographic oh so much, but then I get to tests and think I know EXACTLY where the answer to that question is in the book, the page, the sentance...I just can't remember what the sentance SAYS!) Then the sections order themselves like an album, one track moving seamlessly into the next.