This idiom might be hard to understand if you've never used a record player before.
To put something on some other thing can be used to mean that you physically pick up thing1 and place it on top of thing2. For example, you can put your keys on a table.
The idiom "to put some music on" comes from the time where in order to listen to music, you actually did physically put a record on the turntable of the record player. And before that, on the grammophone.
So, we go from
- to put a record on the record player to
- to put a record on to
- to put some music on to
- to put some (jazz / tunes / sounds / Foo Fighters) on
You could even hear something like to put some Miles on, which would mean that you listen to a recording of Miles Davis.
And even though the days where you physically placed a recording medium onto a player mechanism, the idiom persists to this day, where we typically don't even have physical mediums we store our music on.