Dealing with ANT's is confusing for me also. I did this therapy when I was a lot younger, maybe 15 or more years ago. It all falls into place eventually, on its own, if you practice consistently as advised. A big part of the therapy is to not pressure yourself or try to be perfect with things, that includes how you approach doing the therapy and dealing with ANTs.
The way I deal with it is, I read a few paragraphs from the ANTs handout and then try not to be too vigilant about the ANTs or worry about them too much. I try to catch them sometimes but do it very imperfectly. I don't know if this is the wrong advice, but for me, if I try to catch them too much I can feel really anxious and agitated with racing thoughts. It does make sense eventually, your brain figures out what to do in a natural sort of way, after a lot of practice. Patience is very important with the therapy, and with ourselves in general.
I think the concept is that the ANTs are automatic so we shouldn't beat ourselves up about them and they're negative thoughts so we shouldn't pay too much attention to them because they're skewed negatively so they're not accurate or rational. Trying to 'overcome' them might be the problem. Everyone has negative thoughts. I think the idea is learning to let them pass without engaging with them so much and eventually they'll quiet down and go away. It's like if you pick at a scab, it can last longer and get infected and scar and such, but if you leave it alone and live your life, it will go away on its own.
I'm only 10 months into the therapy. Last time when I did the therapy for a few years, it all fell into place. The reason why I'm doing the therapy all over again is because I really messed myself up for a while and went against everything I taught myself with the therapy. Anyways, I'm on a good track now and still have a hard time but I know the therapy's going to make things easier in the long run. Good luck to both of us