Because his own letter clearly admitted that his behavior toward his wife was wrong and not up to her / his expectations.

It's a PR statement for a country that is generally conservative about these kinds of things. Could be true I suppose.

Why would he apologize if they weee in an open relationship?

Come on, think before you comment

You've not commenting on an apology from him to her. You're commenting on an apology from him to you.

It's a public document; the reasons he published it aren't even likely to include whatever relationship he has with his wife.