These are good questions. I was actually writing a post about this but want to respond briefly...
How can you say you love me when you did this to me?
They can't. Well, let me clarify. I don't think it is possible for a WS to truly love their spouse/family and yet lie, deceive and put themselves first. Love is a complicated topic however. I don't think the problem is always one of "not loving you", it is often more one of "they were not capable of loving you" at the time. The issue for most WS's, when you really get down to the core of things, is that they don't love themselves enough (or at all) to have the critical character traits that prevent other people from doing such damaging things. They don't feel that they are worthy of love, of dignity, of real happiness. The sad truth is that the BS is collateral damage in the bomb that is the WS.
This is a bitter and often impossible pill for the WS to swallow. You, as the BS, see "the real WS" now for who and what they are. You already know there was no way that they loved you and yet cheated on you. But the WS has to face that harsh reality about themselves before they can accept it and change. That is where most WS's get stuck, because accepting that they are so damn broken that they aren't even capable of really loving the people they care most about in this world is just too painful to face. Up comes the denial and defensiveness, the self-pity and the self-destruction.
What I would add however, is that this doesn't mean that there was/is nothing. Love is not an all-or-nothing thing. Your WS may have loved you as best they could, and as much as they could, it is just that their best is based on the empty hole they carry around inside of themselves. In other words, if love were money, you may love your spouse and kids a whole dollar's worth, but they only loved you back a nickel's worth. However, if a nickel is all they had, then that nickel was them giving you all they could, and that's something to consider.
By learning to love themselves, the WS's capacity for true and meaningful love will also grow. Empthy comes back. Compassion. Honesty. Authenticity. If the WS is willing to do the work and capable of seeing themselves as their "true selves", then you can actually build something that was better than before, because your WS will be capable of true love.
How can I prove I love you after I did that to you?
The million dollar question. As you already know, words are meaningless at this point, at least, until backed up with actions. Love is proved through actions and consistency.
Most of the advice in books and even on SI is based around "damage control" and things the BS needs shortly after D-day hits and the BS is still dealing with the shock and pain of what just happened. So things such as NC letters, timelines, transparency, and so on are recommended, however, none of this has anything to do with "proving love". It's just triage.
The WS "proves love" by putting the BS first in all things. By putting more effort into saving the marriage and themselves than they put into having the affair. By being sacrificial in nature and putting the BS first in all things, even to the WS's own detriment if needed. By showing empathy, honesty, understanding, compassion. By dropping their defensiveness and owning the damage they've done. By showing true remorse. All of these things and more are part and parcel of loving someone. You will know love from your WS when you see it, and odds are, it will come in the most unexpected form. But you will know it. They will do something so completely unselfish that they don't even realize it, and that fact that they did it without realizing it, means they meant it.
The biggest thing, in my opinion of course, that a WS can do to show love is to fix whatever it is within them that allowed them to cheat in the first place. Again, they must love themselves first before they are even capable of loving you. The effort to pursue that goal and to make progress in that goal is, in itself, a loving act, and is about the best they can do in their broken state. Fixing themselves makes them a safer partner and a better person, and that alone is a good start to being able to love another unselfishly and authentically.
I hope that makes sense.
