In evolutionary psychology, a trait or predisposition to a behavior are maintained when it provides a benefit at least some of the time. It doesn't have to always be beneficial to be passed on, but it at least can't cause so much harm that a person dies before they are able to have offspring. If a trait is valuable, as in it leads to more viable offspring, then it will be more likely to be passed on to those more numerous offspring. If that trait leads to more offspring, it will be valued by the people, who will seek the trait out in mates. So you have something that's beneficial, and being valued by people, and it will be expressed more than traits that are harmful and therefore not valued. With something like honesty, that's tricky. Integrity and honor make people more devoted and trusting, and having honest leaders who were loved would benefit a society. So one city of honest people would do better, in general, than a city full of dishonest ones (this is a gross simplification, not taking into account any other traits the two people might have.) But in mating, which is a big part of evolutionary psychology, honesty means more than just being a good person. Honesty is desirable in a mate for many reasons. A big one is that men don't have the certainty that women do that they are the true parents of their children. Another is that honesty in a man is a good indicator of how reliable and devoted he is, important things to a woman who invests so much into gestating a child. So honesty gets passed on not just because we desire it to make our societies stronger, but because we look for it in mates to help ensure that our DNA is well cared for.

Humans are very social animals. In our evolutionary history, our lifes depended on the group. We interacted with each other, and for altruism to evolve, there has to be some kind of cheat-detector. Honest individuals that help others will receive help, cheaters that accept the help but don't work themselves will get punished. This started a co-evolution, cheating skills, honesty and cheating detectors evolved. This is one of the main reasons why humans are so intelligent, this 'software' needs a lot of intelligence.

The need to reply on others and their ability to remember you.