My wife and I practice a variation of this. I had no idea it was remotely controversial until today's kerfluffle.
I took it to mean that he really really wants to prove to his wife that he's a trusting, monogamous man.
That implies that Pence feels the need to prove to his wife that he isn't an adulterer by deliberately limiting his contact to other women to the absolutely necessary minimum. Sounds more like the opposite of "trust" to me. (And yes, it also implies that he primarily regards women as "people you can have sex with".)
That's not the reason why Evangelicals promote the practice though. People (both on Atlas and off) have setup this ridiculous strawman that it's because men can't be trusted or something.
The logic behind it is to:
a) Avoid the appearance of evil
and
b) Stop adultery before it starts
Most people aren't going to start sleeping with their staffer the second they're alone. But throw in some late nights working together and a bit of booze, and yeah maybe something might happen. By avoiding these situations you make adultery much less likely, since one's willpower is tested a lot less with the rules, than after you've developed feelings and been drinking.
People apply the same logic to diets all the time, but no one calls you a glutton if you decide not to buy junk food at the grocery store to avoid testing your willpower at home.
I strongly disagree with Pence, obviously—I’m an unmarried straight man with mostly female friends and a virgin at twenty-four—and I think this is a terrible rule for a public figure to have, but I also think it’s silly to descend into histrionics about it being intrinsically sexist for couples to have boundaries like this.
The 'don't judge' crowd seems to be quite judgy about this.