I’m not at all sure that it is snobbery. As I undertstand it, the theory goes something like this:
A large part of the functioning of the human brain is pattern recognition – we actually see and hear much less than we think we do, the gaps being filled in subconciously by our brains, comparing fragmentary sense input with templates built up from our experience. This natural ability to intuit patterns is the basis of our ability with language, mathematics, sports, activities such as chess, and yes, music.
Because of the way our minds work, we are always subconsiously seeking patterns, even when there are none. This is why Percival Lowell saw canals on Mars – his eyes saw faint dots, but his mind connected them into a pattern of straight lines. This is why we see faces in flames, or animals in the shape of clouds.
The brain can be thought of in some ways as analogous to a muscle – in the sense that the more you use it, the stronger it becomes. Give it something find patterns in, and it will work away at it, even without you being aware of it.
The virtue of Classical music (though not solely Classical music) is that it rich, complex, and structured. There are several layers of patterns for the mind to latch onto and “work out” on, even though you are not aware that it’s happening (or even, perhaps that you’re listening to the music at all).
For what it’s worth, I don’t believe that anyone has said that only Classical music can have this effect – I’m sure that many types of Jazz or other popular music would have the same benefit, provided that it was sufficiently complex, structurally.
One other benefit of Classical music, though, which I have seen suggested, is that certain works (I believe it may be by Mozart) have the effect, by reason of their chord structures, tempo etc, of lulling people into a state of calm, alert concentration – ideally suited to learning, or to performing complex tasks.