The idea behind synesthesia is that, basically, one sensory input generates multiple responses. Think of it as, when you hear a note, you might also "see", or perceive, a color related to it. It's not limited exclusively to notes generating colors; one famous example is Daniel Tammett, who was able to recite a very large number of the digits of pi from memory by associating numbers with colors and using the memory of those colors instead of trying to remember the digits.
This phenomenon gives you multiple ways to construct music; instead of just stringing together a sequence of notes and sounds, you can effectively paint a picture or tapestry of color. This also aids, particularly, in note identification. We may have trouble normally distinguishing between two adjacent notes, but it's far easier for the eye to distinguish between two similar shades (say, blue and indigo). So if you hear two notes that evoke different colors, you can better identify specifics.
Synesthesia, basically, gives a greater toolbox to work with. If hearing notes evokes specific stimuli, you can use those stimuli to aid in picking apart music and constructing it yourself. It should be noted that, among people with synesthesia, they tend to have "perfect pitch" at a much higher incidence, which is also linked to musical aptitude.
One final note: synesthesia's effects are not the same among all people with the phenomenon. Someone might see a note as blue that another person sees as fluorescent yellow.