Not the benefit to Earth
The main direct benefit IMHO is not to "Earth as a concept" or a global government or the people back home.
In a realistic scenario, apart of warm fuzzies of exploration, the only positive benefits occur to the actual colonies and to 'humanity' as a whole that includes both Earth and colonies.
Colonists gain opportunities that would be impossible for them on Earth (and would self-select from the subpopulation that desires this tiny set of opportunities specific to space travel). Humanity becomes more resilient, especially to existential threats.
But "government of Earth" ? It would gain some benefits in R&D testing and exploration, absolutely zero benefits in resources obtained, a PR effect that might be positive or negative depending on various random details, a significant cost, and some future political threats, as there are no reasonable ways to control the colonies in the future - it's essentially creating a new civilization that might become cooperative or competitive over time.
There is no direct benefit to anyone staying back home - the invested resources really won't bring any returns back to Earth over their lifetimes, the expansion of the humanity won't give them more space (apart from the minor effect of the people leaving - which could be achieved much simpler by killing them), and the safety of humanity won't prevent their existential threats, rather vice versa - it would make it slightly more likely that Earth gets destroyed, and knowing that humanity lives on elsewhere probably won't make it easier for them when they go down with their planet.
For mankind
The reasons to actually do so are simple - idealistic reasons "this should be done" for the politicians and funders, and personal reasons/ambitions that cause people to do reckless things for glory. History has shown that there always is a significantly large minority of people that find those reasons suitable enough, mostly young males with no commitments (or unwanted commitments), nothing to lose and everything to gain.
Going on an interstellar colonization is practically a far better prospect than volounteering and going off to a far away land to die in someone else's war, and reality shows that there are quite enough people willing to do that. 207