Learn English – ” ‘Low’ back pain” vs. ” ‘Lower’ back pain”

It's common in the context of physical sports to speak of the pain barrier.

If you suffer some kind of injury (on the football pitch, say), your first instinct might be to stop moving so as to protect yourself from further damage. But of course, our bodies and our response patterns have been evolving over millions of years - if the injury was caused by an attack that might still be ongoing, you're more likely to survive (and leave descendants) if you ignore the pain, and keep fighting or fleeing.

It may be that playing through the pain [barrier] will actually cause additional damage which could have been avoided by stopping immediately you became aware of a problem. But just as failure to fight/flee could cause you to die if you'd been injured by an ongoing attack, failing to play on might mean you/your team lose the game.

The sense of through here is to a point beyond pain (rather than continously, as in "I sat through the entire lecture without understanding a word"), but note that normally you're only going through the pain temporarily. In most cases continued activity will cause more pain in total. But for that critical period in the game/fight, your body will produce endorphins (natural painkillers), helping your adrenalin-charged brain to force you to fight/play on. You might well think it that was a good trade-off in the circumstances.

It's the second meaning in use here, that is:

Pass over (a stage or obstacle)

In this case, the "stage or obstacle" is the "pain" involved in the learning curve. The sentence is essentially saying:

It was a lot of work the past year, but, if you learn from what we've learned, it might not necessarily take you a full year to accomplish the same thing.

where the authors have elected to shorten "hassle of learning" with the single word "pain."

The phrase "leapfrog the pain" isn't necessarily idiomatic, but, in this case, pain isn't much different from barriers (at least not syntactically).

That said, the phrase "blood, sweat and tears" is very idiomatic, and it means "a tremendous amount of hard work and effort." It's important in this sentence, because it helps the reader understand what pain is referring to.