Edit: I hate long posts. Marking the important stuff with bold, for emphasis.
While I intellectually understood bet/folding quickly, it took me quite some time to "accept" it, since - on a high level - it seems really weird. I mean, two bets will take us to showdown, but we're willing to put in two bets and then fold?
So why do it? Well, because it brings the highest value against certain ranges. By far.
We're up against a reliable (and/or passive) player. We have JJ. He raised us on the K-high flop and that usually - but not always - means he has a king himself. He sometimes bluffs the flop, so we don't trust him, and we're in position. We take one off. The turn is a rag that puts three to a straight (987) on the board. He checks. We could bet here, but since he's still pretty likely to have AK or KQ, there's just a very good chance that he's trying to checkraise us or otherwise extract more value. Raising the flop and then checking this particular turn is strange.
So we check behind. The river is a jack, giving us a set. He bets out. This is the safest spot to raise/fold, and here's why it makes us money:
Our hand is statistically likely to be better than his. He's a lot more likely to have AK or KQ or even 99 than specifically a ten or KK. A LOT more likely. Raising the river is great, because we do it when we're likely huge favorites.
But if he raises this river, the situation has suddenly deteriorated into going from a clear favorite to a complete loser. We raised the river because we figured to have the best hand (and he's likely to pay us off all the times that we do). Now we have to fold the river because it's painfully obvious that we don't.
Just for shits and giggles, let's try out a few ranges:
70% of the time, he has a lower set, or AK or KQ.
20% of the time, he has us beat.
10% of the time, he's bluffing
(thanks for that nice round number, Mr. Harrington).
70% of the time, we'll make two big bets on the river.
20% of the time, we'll lose two big bets on the river.
10% of the time, we'll win one big bet on the river (he won't 3-bet bluff, like, ever)
EV = 1.1 big bets.
Just calling against the same range of hands: 0.7 - 0.2 + 0.1 = 0.6 big bets.
Now, since someone will (or should) argue that we don't have that perfect read on someone. There HAS to be some times that he'll 3-ball this river as a bluff. How much will folding cost us if we have a gutsy enough opponent who will try that half the time he's bluffing?
EV = 1.4 - 0.5 + 0.1 = 1.0 big bets. Still much higher than calling.
Now, for bet/folding or raise/folding to be correct, these two things have to be true:
1. You need to have the best hand most of the time, and your opponent must be willing to call with a worse one.
2. The board (or the opponent, or both) must be such that the likelyhood of being raised with a worse hand is very small.
#1 is the reason we're raising. #2 is the reason we're content to fold if we get raised.
You can get by very well without ever bothering with bet/fold or raise/fold lines, but as you can see, they do add some profits in marginal spots. Probably not more than a fraction of a BB per hundred, but it's there.