Any drummers out there? Why do drummers hold on to the cymbals when striking them with their sticks?

It's called a choke. It's a specific technique used by drummers, not unlike a palm mute on guitar. I've been a drummer for almost 20 years, and honestly it's not something we do all that much.

They don't always do that. I'm no drummer myself, but I would think that it's because they don't want the sound to go on and on if it's not supposed to in the song. Holding on to it would mute the cymbal and stop the sound quickly.

It shortens the sustain. When you hit a cymbal it will ring for a little while. To get a short, quick cymbal note the drummer hits the cymbal then quickly silences it by grabbing it with his hand.

When they grab the cymbal with their hand it's because they don't want it to ring out, so it's about altering the sound it makes.

If they didn't grab it the cymbal would keep ringing during a part when it's supposed to be silent.

alter the sound