Apologizing is one of the hardest things we have to do, not just as leaders, but as people.
Telling someone you've done them wrong is never easy. When we do finally work up the courage and humility we need to apologize, we tend to make excuses to save face. Excuses may make us feel better about our apologies but they take away from the sincerity of our message.
If you're in a position of authority, formal or informal, you may need to raise your standards. Now. The hard truth is the only acceptable apology is the excuse-free apology. Ban the words "if I offended anyone"!
Let the Light In
People who aren't trustworthy try to create murkiness, doubt, and unnecessary mystery. They construct a narrative of what happened to minimize their mistakes, at least in their own mind. People who can be trusted, on the other hand, deliberately create transparency as often as possible--in how they make decisions, what the criteria for success is, how they assign projects, and more. This same principle of transparency applies to apologizing: shine light on what happened and your role in it.
It sounds terrifying to bring your missteps into the open--but on the receiving end of a sincere apology, most people are appreciative and often genuinely forgiving. A good apology can even turn your detractor into a defender in the future.
Make an Honest Effort to Change
Many people apologize but don't change the behavior that precipitated the apology. You can't talk your way out of a problem you behaved yourself into you. You can only behave your way out of it. If you want someone to believe your apology, you're going to have to change your ways.
You may even need to offer some type of restitution. This isn't necessarily financial and could be as simple as making an apology in the same forum in which the offense occurred. Did you repeatedly speak over a team member in a weekly meeting? Apologize for that behavior at the beginning of the next meeting and stop speaking over your team. If your offense was an interpersonal slight, take the person you offended aside, look them in the eye, and offer an excuse-free apology.
Don't Follow the Pack
Nowadays, the go-to approach for recovering from a mistake seems to be to admit nothing, deny everything, blame other people, and come up with an alternative narrative. But these tactics are simply manipulative: the real goal is to turn the facts on the other person and make them look more like an instigator than the victim.
This is toxic leadership. It is the polar opposite of principled, trustworthy behavior and an example of how to torpedo a relationship. If you want to rebuild a relationship, you'll need to act in a much rarer way: by taking responsibility for your actions and becoming comfortable admitting you're wrong.