I can see the neighbors yard slopes down towards your yard, combined with a (hopefully) slope from your house towards the fence, so the fenceline is always going to be the collection point for water, whether a fence exists there or not. If there is more pooling now, probably what happened is the spoils from digging the holes was just thrown down along the fence and messed up any (if at all) slope to a lower area that existed before.

The first solution would be to create a swale that goes somewhere lower, looks like maybe beyond the back property line? So you would grade a slight "dip," either on the neighbors side (if room) or your side to direct water to move somewhere lower.

The next step if that is not possible or sufficient is to install a french drain, which would collect water underground and cart it off to somewhere lower. That discharge point would need to be lower than the depth of the french drain, so that could be a limiting factor in it's use as a solution.

Thank you so much for taking the time to respond. I have a company that does these types of things coming this week. I'll update when I hear but I think we are on the right track! (Prob 15:22- Plans fail for lack of counsel, but with many advisers they succeed)