When I was even younger I would say I'm sorry more times then I could count. I'm sorry I said that. I'm sorry I did that. I'm sorry the traffic sucked. Yada yada yada. Though I started to learn this pre-TRP, I put it to more and more practice and have definitely seen results:

Stop saying "I'm sorry" for shit

Now it's okay to admit when you're wrong, that's normal and the right thing to do. But, saying you're sorry doesn't change anything. The "cliche" example is, throw a plate on the ground and watch it shatter. Say you're sorry to it. Did it fix itself? It's cliche but it's true. You fucked something up at work? Did saying you're sorry fix it? No. Did saying you're sorry fix the fact that you had one too many beers and fucked some shit up with your friend? No. People hear "I'm sorry" so many times and people apologize over the smallest shit (much of the time they don't seem to mean it either), that it just becomes a useless phrase. People hear you talk about how sorry you are for the 10th time, they don't really think you're that "sorry" for it because you keep having to say it. Additionally what does that really mean? If I just say I'm sorry then everything goes back to normal! Cool No, that's not how it works.

People don't care if you're sorry, they care what you can/will do about it

Let's say you fuck up at work, don't tell your boss you're sorry, tell them how you're going to fix it. That's the key. Don't explain how because of X, Y and Z it's screwed up. Show them how you're going to remedy the problem. For example, if I fuck up somebody's order at work I don't tell my boss how sorry I am for screwing it up and how it won't happen again. I grab it, fix it and hand it back to him as quickly as I can, making sure in the process nothing fucks up again. Then, in the future, I make sure it doesn't happen again. I don't need to tell him how it won't happen again. I show him by not screwing up. People will react more positively to you and from my experience demonstrate more respect for the people who don't waste their time apologizing but instead acknowledge their failure/ fuckup and instantly move towards resolving it.

Another smaller example I've seen is changing "[I'm] sorry" to "Excuse me" when just trying to pass through people. Simple but effective.

On the flip side, don't let people get off the hook for just saying "I'm sorry" to you

If they're actually sorry, they'll fix it. Otherwise they'll just say it, which is a waste of their time and yours. If they're not willing to fix it, then they clearly don't actually give a shit about you/respect you enough, which means you don't have much of a reason to be around them.