Mankind wasn't actually driven to extinction.
- Rather, the tale Grandpa Squirrel tells was either embellished by him to drive home the anti-war message, or what they saw happening was just from their perception of what little they know; what actually happened was that the humans they saw were involved in a regional battle or were in a small part of a bigger war that ended up killing many or all of the local people or caused the town that the animals eventually inhabit and rebuilt over to be abandoned. There can also be evidenced by what happens on-screen, since it would be utterly implausible for mankind to have completely killed all of each other off using mere firearms of cruder weaponry of the day (a plot flaw that was addressed with the rise of Nuclear warfare in the 1950's remake "Good Will To Men"). This does bring up the question of why the animals wouldn't be aware of the existence of other humans then, assuming they interact with other animals outside their territory.
It's why the animals believe mankind went extinct. The only humans who knew about the village killed each other — therefore they've never seen any other humans since.
Which is why mankind in this cartoon was so obsessed over killing each other that they brought every reason to war to the point they killed each other with WWI weaponry.
Well, in the Peace on Earth short. The war was probably very long, and, given that rodents don't live long, Grandpa Squirrel was born two weeks or so before the last soldiers finally killed each other.