By apologizing, you take the shame of your offense and redirect yourself. You admit to hurting someone or diminishing someone and in effect, say that you are the one who is disminished. "Im the one who was wrong, mistaken, insesitive or stupid. In acknowledging your shame you give the offended the power to forgive.
Some people apologize to feel better because they feel bad of what they did. Sometimes they just apologize to feel better. A good apology also has to make you suffer, you have to express genuine, soul searching regret for your apology to be taken as sincere.
Do you think an apoligy could change the relationship between two people?