Strictly speaking, Platonism is any philosophical idea or ideology that derives from the teachings of Plato, the ancient philosopher. However, it has also become a term used to define certain aspects of music, and is a source of controversy.
Platonism in music defines the idea that music is in itself a constant, existing outside of the physical realm of notation and performance. While said performance is, in fact, a Realist notion -- it is something in the real world experienced by real people -- the music itself is discovered rather than "created." Not to say that composers do not compose, but they are guided by eternal principles in coming to the proper arrangement rather than by their own originality. The idea is simply that the "correct" arrangement of music is already there, and the writing of it by a physical entity is just a pathway for the music to arrive at the ears. A lesser composer cannot come to the "correct" arrangement because his principles are not properly trained; a greater composer can come to the "correct" arrangement because his principles are properly trained.
Naturally, this view is controversial, and is strongly contested by various scholars. Peter Kivy, professor of philosophy at Rutgers University, is a leading defender of the concept, and he has authored several articles on the topic.