The best way to think about it is that the button presses are an additional clap pattern on top of the song. If you play it exact, you'll get a suitable clapclap-clap or whatever above whatever song you're playing to.
This was really helpful advice. I don't play too many rhythm games so my brain was getting confused about the timing the notes are going for. Really feeling it now.
Favorite track to play/listen to so far is Now I Know (Yuu Miyake Remix).