Don't think of it as an adjective, but more like a verb. It is a verb that is being used like an adjective. In this case, "her offending remarks" means, "her remarks which offend". They are the remarks that offend people.
This almost means the same thing as "offensive remarks", but there's a small difference: If you make an offensive remark, that just means that the nature of the remark is offensive, but it doesn't mean anybody actually got offended. If you make an offending remark, that does mean that the remark offended somebody.
Here's another thing though about that sentence, and it's kind of subtle: There are two kinds of "remarks" or "statements" that it's talking about. The first kind is the "offending remarks". The second kind is the "subsequent apology statement".
By mentioning these two kinds of "remarks", "statements", or whatever you want to call them, the author is trying to say something about her: The author is saying that she will say something mean, then she will apologize about it, and she keeps doing this over and over. The author is trying to show a contrast here: First the mean things she says, then the apology she gives later. (It doesn't sound like the author thinks her apologies are genuine.)
Why does she apologize? It is because her remarks weren't just offensive; they actually did offend somebody (according to the author).