How can atheists be so certain that God doesn't exist?
When theists ask how and why atheists can be certain that no gods exist, they do so under the mistaken assumption that all atheists deny the existence or possible existence of any gods and that such denial is based upon certainty. Although this is true of some atheists, it is not true of all indeed, it seems unlikely that it is true of most or even a significant minority of atheists. Not all atheists deny the existence of all gods and not all of those who do claim absolute certainty.
So, the first thing to understand is that atheism is simply a matter of lacking belief in the existence of gods. An atheist might go further and deny the existence of some, many, or all gods, but this isn't necessary for the "atheist" label to apply. Whether or not an atheist does go that extra step with regards to any particular god depends upon how "god" is defined. Some definitions are too vague or incoherent to reasonably deny or affirm; others are clear enough that denial is not only possible, but necessary.
The same is true for whether or not an atheist claims to be certain in their denial of the existence of any gods. Certainty is a pretty big word and many atheists consciously model their approach to the existence of gods on the naturalistic, skeptical methodology of science where "certainty" is typically avoided except where it is unquestionably warranted. In science, belief is proportioned to the evidence and every conclusion is considered basically provisional because new evidence in the future might, in theory, force us to change our beliefs.
If an atheist is going to claim certainty in their denial of the existence of gods, it will often be because there is no logically possible evidence that could force a change in their conclusions. It may, however, also simply be a position based on probability: in the world outside of science, most people are willing to claim "certainty" if contrary evidence is extremely unlikely and not just impossible. Either way, though, the definition which a theist uses for "god" will play a critical role in what sorts of conclusions and certainty an atheist is likely to draw.
Some theists define their god in a way which is logically contradictory much like claiming that their god is a "square circle." Square circles cannot exist because they are logically impossible. If a god is defined in a way which is logically impossible, then we can say "this god does not exist" with a great deal of certainty. There is no way that we will ever come across evidence which points to the reality of something that is logically impossible or impossible by definition.
Other people define their god in such a way that it is, quite frankly, impossible to understand. Terms used are too vague to pin down and concepts used don't seem to go anywhere. Indeed, sometimes this incomprehensibility is touted as a specific quality and maybe even as an advantage. In such situations, it just isn't possible to adopt a rational belief in such a god. As defined, at least, such a god might be denied with some certainty because the chances of having evidence pointing to an incomprehensible god are pretty low. Most atheists, though, will simply refuse to believe or deny such gods.
So, how can atheists be certain that no gods exist? A person doesn't have to be certain of the nonexistence of gods in order to be an atheist, but just as important is the fact that most people aren't absolutely certain of many of the things they believe or disbelieve. We don't have perfect and irrefutable proof of most things in our lives, but that doesn't stop us from navigating the world as best we can.
A person doesn't need absolute and perfect certainty in order to be either an atheist or a theist. What should be required, however, are very good reasons for whatever direction a person goes. For atheists, those reasons are at the very least the failure of theists to make a good enough case for either theism generally or any specific form of theism to warrant adoption.
Theists on the whole think that they have good reasons for their beliefs, but I have yet to encounter an alleged god which warrants my belief. I don't have to be certain that those claimed gods don't exist in order to be an atheist, all I need is to lack good reasons to bother believing. Perhaps someday that will change, but I've been at this long enough that I rather doubt it will.