… when someone has transgressed a norm we become rightfully wary that they may do it again. The power of apologies, according to Martin, lies in the fact that they help to mitigate this threat of future transgression. They indicate that the person recognises the transgression and is willing to take steps to make sure it does not happen again. [...]
You must make some effort to change yourself (or your organisation) in order to ensure that the same thing does not happen again. The artful way to do this is to demonstrate change, i.e. to show that you are actually doing something. The non-artful way is to be complacent, i.e. to not demonstrate any changes.
If your apology doesn’t include an explanation of why you wronged and how you are not going to do it again, then it’s likely to be ineffective, since the other person has no reason to assume that you even understood what went wrong, nor that you know what you need to do in order to avoid repeating the mistake in the future. (As that linked article notes, there is a difference between an apology and an excuse in general, but here you seem to be using “excuse” as a synonym for “explanation”.)
The problem with this rule is that a decent apology is commonly seen as including an explanation:
If your apology doesn’t include an explanation of why you wronged and how you are not going to do it again, then it’s likely to be ineffective, since the other person has no reason to assume that you even understood what went wrong, nor that you know what you need to do in order to avoid repeating the mistake in the future. (As that linked article notes, there is a difference between an apology and an excuse in general, but here you seem to be using “excuse” as a synonym for “explanation”.)
Go read Aaron Lazare “on apology” for a more complete picture of apology.