A common interpretation is an admittance of wrongdoing. That’s a heavy blow to the ego to the person saying it, especially after a heated disagreement. Which is why the phrase is so hard to verbalize.
But what if we see “I’m sorry” as a way of communicating that we want things to be better, and are ready to move on?
Spoken this way, more often than not, they end up making that very intention come true. If that’s the outcome that matters, it’s probably worth a shot.