You read that correctly. When you apologize for some action or inaction there may well be a reason. If you explain that reason as part of the apology you’re making an excuse, not making an honest apology.
An easy example is when you’re late to pick someone up. The reasons could be the drive was longer than you thought, you lost track of time and left late, you were stuck in traffic, or some similar reason. Of course there was a reason or you would have been on time (or even early). Does it matter why? Not when it comes to saying you’re sorry to someone for being late. By explaining yourself instead of just apologizing you cheapen your apology. The person may ask why you were late. If so you can decide whether to explain yourself or just say “it doesn’t matter, again, I’m sorry I was late.”
Many people offer the reason/excuse first and then apologize. This is no better than providing it after apologizing. It may be worse too. Let your apology stand alone. It means something because this is about you making an honest apology, not how it’s received.
The person you apologize to may not acknowledge your apology or accept it. All you can do is put it out there. Once you do that there is no guaranty of anything other than you put it out there with sincerity, intent, and no excuse. The next time there is reason for you to apologize to someone, do it and see how it feels.