As a necessary way of being around others. You're going to fuck up, others are going to fuck up, and you have to make things right. Apologizing is cheaper than many other ways of making things right.
How do you do it?
A few ways.
My first way I learned was by doing something. I didn't explicitly say sorry, I just.. did a thing to fix it somehow. I learned this from my dad, and that's how he does things. It isn't very efficient, and I do a lot of work only to discover that fixing someone's thing does not equate to fixing the other thing.
Then I have my... work, professional, indirect sorry. "Thank you for understanding." "I appreciate your waiting, I know it took a while." "I was rushed earlier, but I'm sorry if I came across as brash earlier." It isn't quite a true feeling, emotional sorry.. so I have no trouble saying it.
Then the minor misunderstand apologies. "Oh, I'm sorry.. I thought you said 3:00! Oops!" "Oh, I didn't see you there!" The ditzy things that don't really matter. Again, I don't feel much genuine emotion behind it, it's more a communication tactic. I acknowledge I got a wrong message, while understanding communication works as a two-lane road. So, a sorry is okay here.
Then.. Just apologizing for something I did wrong. Really sucks. Like. Super sucks. I beat the shit out of myself.. and the apology just stands for all of the awful things I'll say/think to myself for fucking up badly enough to truly apologize. "I invalidated the way you feel, which is something I despise in others, and I hurt your feelings in the process which feels just awful.. On top of that, you'll be less likely to confide in me about things later. I'm so so sorry." I just can't .. do it right away. I get angry--at the situation, at myself, at everything. I can get there.. I'm getting better about it slowly. But I still can't come to that acknowledgment right away. Ego-protecting bullshit in the way there. So, I have to get angry, then super depressed and emo, and then apologize and feel shitty for a while.
How do you value the act?
Apologizing is the best thing you can do for yourself and others. Humbling yourself to truly apologizing, and meaning that apology.. it's an incredibly hard thing to do. Not the bullshit "oops!" sorrys.. but a true, genuine I-fucked-up-and-just-saying-that-isn't-enough apology. It's challenging, and it tests all of your own biases and thoughts. Which, of course, means I value it highly yet despise it in the immediate act. I always feel better (eventually, after I get over the emo part of it) after I've done it.
And it can be sooo hard. Apologizing to a person is SO easy to not do--pawn it off on any number of stupid illogical emotional biases. "Oh he was a jerk once, so I'll just take off a jerk point (but not really because I gave him a million)." "Oh, she didn't really care anyways so I won't really bother." "Oh, they know that's how I am and how I do things, so I don't need to say them words out loud they know how I do. All sorry like and all." "Well, they were a jerk last week, so I'll stop being angry about that and decide not to apologize and communicate nothing of this with them." "How can they expect me to be sorry about that?! It was out of my hands! Sure I could have done something but hindsight is 20/20."
There's a lot of garbage people tend to throw at apologies. But they're super important. Knowing when you truly fuck up and acknowledging that to others is probably an important way to not be a fuck up in the future and also keep relationships together.