From Doc Wong:
To apologise is to express our recognition of and sorrow for having hurt another.
Giving a sincere apology, without reservation or self-justification, is one of the forms of generosity. By doing so we help heal any anger or resentment the other person may feel, we ease the way for them to practise forgiveness and we make possible the mending of a ruptured relationship.
On our part, giving a sincere, unreserved and timely apology soothes any self-reproach we might feel and helps us become more open about and objective towards the negative side of our character, which is an important part of character building.