From my understanding of this messed up situation:
She has half of the equity in the house, he has the other half, maybe not exactly half and half, this is common after a divorce and the actual proportions are irrelevant. In order for HER to acquire full equity in the house (the house she lives in and considers hers and that the divorce has apparently assigned to her), she must pay his half of the equity BACK to him when she refinances. Thus, she both "has" equity and "owes" equity. Both are true. She has her half (which is money on paper and represents her ownership of the house, not money you can spend), she owes the other half which is also equity, just not hers (which is real money she DOES need to spend, and will be refinanced into the form of a larger mortgage with higher payments). Once she refinances, he gets the lump sum representing his equity, and she gets a bigger mortgage. He could then use that lump sum to pay back the alimony with. But that's all in the future, and the future is an uncertain place that doesn't help anyone now and that nobody wants to wait for.
So she suggests that she'll forgive his alimony if he forgives the equity she owes him, because that is probably a much bigger amount of money that she owes him and would then get to keep, compared to many many years of alimony that she might not even get if he's not going to be around that long and isn't working or goes bankrupt or whatever else might happen in the intervening years.
The key moment in the video as far as I'm concerned is when she mentions her husbands "new wife and kids". I think that if you strip away all the reasonable-sounding explaining and arguments, that's what this is really about, she wants to get as much as she can in cold hard cash right away, even if it means cashing out some of her ex-husbands 401k now, so that the other wife and kids don't get their hands on it and then she doesn't have to worry about them anymore.