If mankind stays bound to a single planet then it is ultimately doomed.

It's demise might come quickly due to global warming, nuclear war, a global plague or a million other causes we are not even thinking about right now.
Looking at longer timeframes there is the danger of a cosmic impact, natural changes in the environment (ice age,...),...
And even if we deal with all these we will still be gone in about 10^9 years because the sun is slowly moving towards the end of its lifecycle.

By moving to other planets and having them become self-sufficient we can deal with a lot of these issues and extend the lifespan of the human species (or its evolutionary descendants) with millions or even billions of years.

And once we have become a truly interstellar civilization (or even an intergalactic one) we can start playing the long game and prepare to deal with the eventual heath death of the universe; extracting energy from black holes, building outposts around dwarf stars,...

A dedicated civilization with proper long term vision could potentially last for many billions of years ones it has a truly interstellar domain under its control.
But that would require leaders who think beyond the next series of elections...