Integrity is a good moral value and something that will help people throughout their lives. Your question appears to be asking only about how integrity ends up being helpful to people, not why it is morally right. So I will address that issue only.
Integrity helps us be trusted by other people. This is absolutely vital if we are to have good personal relationships and if we are to succeed in any sort of a meaningful career. In personal relationships, integrity is key. Imagine being married to someone who had no integrity. You would never be able to trust them to do the right things financially, or with your children, or in terms of being faithful to you. At work, integrity is key as well. No one would ever knowingly entrust responsibility to someone without integrity. Someone without integrity would, for example, be likely to steal from their employer or to do as little work as possible instead of trying hard. Neither of these is a trait an employer would want.
Integrity is about what you do and how you act when no one is watching. By teaching children the importance of integrity, they are learning that doing the right thing is not only important when someone else witnesses the act or behavior. If everyone acted with integrity, the world would be a much more trustworthy place.