Contribution is earned by defeating demons and completing crow missions.
As of now, contribution has only two purposes, exchanging it for Muzan blood (100 contribution for 2 blood) and exchanging it for money (how much money you get depends on how much contribution you have e.g. 1 contribution = 1 money. 69420 contribution = 69420 money)
The person who exchanges contribution can be found at the entrance to the training area which is also where you may find the moon breathing trainer.