Why would sex be a way to reduce anxiety? Why would masturbation be a way to reduce anxiety? Why would self harm be a way to reduce anxiety? Why would bingeing and purging be a way to reduce anxiety?
There are a ton of different harmful behaviors that reduce anxiety. I don't know why it is.
When you write "why does an abuser abuse out of anxiousness" the logical answer is "to reduce his anxiety". He has a feeling and he responds to it - with no thought to anyone or anything else except his own immediate feelings.
Why he is anxious and why he ignores the legal and moral implications of his actions are different questions entirely. The human drive to reduce or avoid negative feelings is a powerful one.