Why do theists think threatening atheists with punishment in the afterlife will work when we don't even believe in an afterlife?

That makes as much sense as asking why a DANGER - HIGH VOLTAGE sign should mean anything if I don't believe in electricity.

Wake up !!!!!!!!!!!!

Why are atheists unable to distinguish between a threat and a warning???

As a lifelong theist (Protestant Christian), I agree. It's really stupid to threaten people with some punishment they don't believe is real. It's also pretty arrogant, because the threat is always for not believing THEM. That is, not believing what they say; they rarely care whether you believe in their God. 

Personally, I do hope for an eternal life, but I rather expect that whatever that turns out to mean--I don't trust anyone who claims to know all the details--it will also be enjoyed by lots of people who don't hold the same beliefs as me. I figure anyone who presumes to set limits on the redeeming grace of God (assuming, as I do, that God and his grace exist) probably underestimates them. 

But I actually decided to write an answer here because you seriously misunderstand the meaning of faith. (That's probably our fault, for not being clear about it--and I'm worried that a lot of my fellow Christians also misunderstand it now.) Faith is not any sort of belief; faith is a commitment to a relationship. We speak of "keeping faith" with a partner in an agreement, or being "faithful" to a spouse. It's about trust, not about insisting we're right about everything we believe. 

As for belief, I doubt anyone, anywhere, has ever believed anything without evidence. What we don't have--what makes it a belief--is the lack of proof. Beliefs don't make sense if we have proof, because then we simply have facts. But beliefs, working assumptions, are a thoroughly practical approach to the simple fact that there's quite a bit of reality we can't prove, starting with the basic question of what actions are good and what are bad. (The whole philosophical field of ethics is about posing, comparing, and arguing over answers, but there's no proof involved.) 

To put this on a personal level: I'm reasonably sure I can't be entirely right in everything I believe. Nobody else can possibly hold EXACTLY the same beliefs as me; even if they use the same words to express them, the beliefs themselves would differ, because words aren't that precise. So it's ridiculously unlikely that I'm right and everyone else is wrong. 

But that's where faith comes in: I trust something greater than myself--the one I call God--to help me deal with the consequences of my being wrong. I trust that I won't be expected to be entirely right, because that's more than I or anyone else could handle. 

So it's not hard to believe that God might help a lot of other people in the same way, and might even think the truths they perceive and I do not are just as important as the ones I perceive and they don't.

They don't have any real arguments, no proof God exists, nothing.  So they resort to the fear-mongering they encountered as children, thinking its logical or at least convincing.

It's charitable to warn those who are about to drive off a cliff, even if they don't believe in cliffs.

It’s mistranslation of just a few words that cause folks to believe what’s called the ‘eternal torment’ gospel. But Jesus said “these (not yet saved) will rise into aionian-kolasis *AGE*LASTING-CORRECTION” (translated, everlasting punishment, oye) “but the righteous into aionian life” (into “the thousand years/millennium age) - Mt.25:46.

Because when you don't have facts and reason to back up your argument all you have left is threats. 

You are completely right.  Although I am not an atheist, I see no proof that there is an afterlife: it is just something that clergy of many religions have concocted to keep their flock from deviating from their religious faiths.  (Preachers make a living by promoting their religion: if members leave their congregations, preachers would become unemployed.)

Religion is faith.  It is not fact.  If it were fact, it would be scientific.

Sounds like someone got TRIGGERED!