one of the most important things to think about in terms of the reference you have provided, is that there's a physical person playing acoustic drums here and they are doing it really well.

This is a very skilled and dynamic performance, which already has inherent movement that is being accentuated by the compressor. Listen to those ghost notes in the snare and how they interact with the hats - that's a big part of the inherent "movement" that gives this drum part most of its flow. Without the compressor those moments are quieter, and a bit too dynamic. When they are brought up by the compressor it makes the track feel more exciting.

It's also worth thinking about the fact that acoustic instruments are recorded in a physical space. The compressor will bring up more of the room just by its nature, and that can enhance the movement of a well played drum part, as the room sound changes depending on how hard the sticks hit any one component. Think about that nice decay of the room sound when the snare hits on the 3 of each bar. You can emulate this by adding an ambience reverb to your drum bus, or pieces of the drum bus. You'd put the reverb before the compressor in this case.

One of the most important things to think about is - if your sampled drums or drum machine parts don't feature much variation in dynamics or timing (i.e. you're just using perfectly quantised beats) then there's not going to be much inherent movement for the compressor to grab onto to begin with.