The smell of spoiled milk has caused the kingdom to be overrun with rats. In order to solve the problem, King Pyro has offered a reward for those citizens who can catch these rats.
If there is a next phase of the Cheesyverse, it would be Phase 3. There is no guarantee that this phase will come, but if enough people start collecting cheese and leveling participating in Phase 2 then there is a good chance that it will.
The gameplay for Phase 3 will be inspired by MouseHunt. This is a game that I played many years ago, and is apparently still going strong. The design for the Rat Traps in the Cheesyverse is very similar to something I remember from MouseHunt.
The idea for Phase 3 is to get some decent art made for various types of rats, and let Trappers use their cheese to try and catch these rats. The endgame has not been completely decided yet, but a race to "catch them all" is one idea.
There is also the possibility of expanding the gameplay to feature more trap types, similar to the way MouseHunt operates. However, all aspects of Phase 3 are unconfirmed.