This is an achievement for using 5000 bombs in Bomber Royale, which can be accessed both in the Geode hub, and also through the menu when in-game. It's like your typical Battle Royale mode but you only throws bombs and there's a few powerups to get. There's not much to say about about earning this other than getting the powerups that reduce the energy cost of the bomb and spamming bomb throws. I'd try to stay away from everyone else so you can use as many bombs as possible before you get killed. It will take quite some time to throw 5000 bombs, as you can only throw about 3-4 before your energy is gone and you have to wait for it to come back.
The real reason I made this solution though is to pass on that it's apparently buggy. I got this achievement while going through the shadow tower. I wasn't using bombs. When it popped I was using the ult of the gunslinger. The only thing I can possibly think of is that the GS ult shots are coded as bomb explosions since they are AoE. Though even if that is the case why it counted outside of BR is beyond me. Maybe it only counts shots in the tower? Maybe Boomeranger can get this with his bombs? Not sure. Anyway, just thought I'd pass it on in case it helps anyone else avoid another shit BR mode.
The new update Crystal Combat dropped, and the patch notes mentioned that there was a bug where bombs thrown outside of BR would count, and that they fixed it. Looks like you're stuck throwing bombs in BR until this pops.ikilledbecka
commented that he got the achievement in the shadow tower, so apparently the bug is not fixed. Based on the comments I assume it counts "explosions" outside of BR aka the gunslinger ult, the fae trickster's bomb, etc.