Why do people need an apology?

I doubt that you can get a clear and specific answer here that will apply to all or even most cases. You might find that the best sort of "answer" available will arise from thinking about the function(s) that such demands for apology play in groups of humans. For example, it's easy enough to see that these are attempts to police social behavior of others. Which is the same function as encouraging/demanding that persons feel guilt about X or Y.

An apology is a sincere acceptance of blame, a recognition of the harm done and a request for forgiveness. If someone harmed you and they refuse to acknowledge it and accept they did harm, it's not over. That's the important part, you haven't stopped harming someone until you admit you were doing it.

I'd take issue with one little thing there

The way I'd think about this is that where there is no (genuine) apology, then a repetition of the act is quite likely to recur.

Some people WANT an apology, but will never get it. They have to learn to accept that.

Some people give a half- hearted apology, but that’s insincere and doesn’t satisfy the harmed.

Some actions like betrayal, abuse, theft, and cheating can cut so deep, that an apology means nothing and changes nothing.

Time and earthworms cures all, they say.

Others trust Karma.

There are many different types of apology... they can be done out of shame, or regret, or empathy, or annoyance, or humiliation, or political strategy

There are some apologies I have received that were quite meaningful to me. Particularly when someone I care about hurts me, an apology let's me know that they value me as a person. Likewise when I hurt someone I care about, I want to let them know that they are important and that I care about the relationship.

The current ritual of public shaming followed by forced apology that is then rejected bothers me. This is more about demanding complete political capitulation then any attempt at bringing about healing or human connection.

When the angry mob comes for their apology all you can do is tell them what they want to hear and hope for the best. I don't find that particularly useful.

To add to this - it also showing that this someone is showing that they care and respect you. For example if it is a friend or family member that hurt you some way - it is showing that this friend and family member shows they care and respects you. That is a big part of how it stops the harming.