Why does “His to lose” mean “He will (most) likely win?

Solution 1:

Romney is currently the frontrunner for the GOP campaign, if we ignore Cain for the moment (who has other matters to deal with). He is doing so well in the race to be nominated by the GOP that many people think he has won, even though the race isn't quite over. In other words, it's a foregone conclusion (at least for those people) that Romney has won the nomination.

Now let's liken this to a different situation: Assume you and I were trying to get an apple, and you had gotten it. It's your apple to lose now, because you already have it, and thus you can lose it. I can't lose the apple since I don't have it, although I could lose the contest as a whole.

So now that Romney essentially has the nomination, it is his nomination to lose; while all but one candidate will lose the nomination in the sense that they will not win it, Romney's loss (if he does so happen to lose) would be a loss in the sense of losing something already possessed.

As for popularity of the phrase, I wouldn't say it's obscure, but neither is it commonly used.

Solution 2:

The phrase "yours to lose" means nearly the same thing as "yours to win". These phrases mean that whether you win or lose from this point onwards depends on your actions, and not on circumstances beyond your control.