Once you get the "ears" past the edge, just slide the tray the rest of the way out.
Note the battery type carefully: It is a ML2032, which is rechargeable. It is NOT the same as a CR2032, which is not rechargeable. Putting a CR2032 in here is a recipe for bad things to happen.
Note the proper orientation (see picture), positive + symbol facing towards bottom. Negative side facing keys. Slide the tray with the new battery back in. The battery can be oriented incorrectly and NOT work.
If you want to make it easier to remove the battery the next time, you could trim the ears slightly. Or slice some notches in the plastic "springs" to weaken a little.