Paraphrase: Someone that has already done you a favor will be more likely to do you a favor than someone you did a favor for.
We tend to like people more once we have done them a favor, because normal people will justify to themselves that they did the favor because they liked the person. Conversely, a villain will come to hate his victim and possibly de-humanize them, as this makes it easier on their mind to kill or ruin the object of their hate.
The reason behind this is cognitive dissonance: an uncomfortable feeling caused by holding conflicting ideas simultaneously. In other words, in order to do positive things for people when you don't necessarily like them, the person doing the good deed has to change their mental thoughts and feelings about the person they helped, in order to ease their cognitive dissonance.
A classic example from literature is from Aesop's fables. In The Fox and the Grapes, a fox sees some high-hanging grapes and wishes to eat them. When the fox is unable to think of a way to reach them, he surmises that the grapes are probably not worth eating, as they must not be ripe or that they are sour (where we get the expression "sour grapes" from). This criticizing is to reduce his cognitive dissonance.
How can you use it to your advantage in writing? A few different ways:
1) Have a wary heroine do a favor for the hero that was imposed upon her (like at church or from some other social necessity). Just by doing the favor, she'll be less disposed to dislike him, even if the hero didn't ask her to do the favor. (But it will be particularly strong if you have the hero ask the heroine for a favor in front of someone the heroine feels obligated to be on her best interest for!)
2) Have a hero decrease his cognitive dissonance after desiring a date with the heroine, asking her out and getting rejected, and then have him start to postulate about what all misfortunes he has likely avoided by not going out with her. (Substitute anything for the woman: a new job, car, suit, new restaurant booked solid, etc).
3) Have your villain increasingly think of the protagonist as an object, dehumanizing him or her. it makes it easier for the villainous acts to escalate (and for the tension to ratchet up in your book) if you do so.
4) The Foot-in-the-Door technique is a modern example of the Ben Franklin effect, and you could use it as a great secondary plot in a book...and bring in some comic relief if your writing leans that direction. Think about it: your do-gooder heroine gets a phone call to ask if she'd sign a petition about a cause. She says yes, just wanting to get off the phone. Then they show up at her door, and because she said yes on the phone, she goes ahead and signs the petition. Then a week later they ask her to donate to the cause she so obviously shares with them. Well, geez! She goes and gets her purse. Next month, they have her knocking on doors for the "cause" and she's wondering, "How did I ever get into this?"
Have fun with this!