"But isn't it impossible to prove the non-existence of something?"

There are many counter-examples to such a statement. For example, it is quite simple to prove that there does not exist a prime number larger than all other prime numbers. Of course, this deals with well-defined objects obeying well-defined rules. Whether Gods or universes are similarly well-defined is a matter for debate.

However, assuming for the moment that the existence of a God is not provably impossible, there are still subtle reasons for assuming the non-existence of God. If we assume that something does not exist, it is always possible to show that this assumption is invalid by finding a single counter-example.

If on the other hand we assume that something does exist, and if the thing in question is not provably impossible, showing that the assumption is invalid may require an exhaustive search of all possible places where such a thing might be found, to show that it isn't there. Such an exhaustive search is often impractical or impossible. There is no such problem with largest primes, because we can prove that they don't exist.

Therefore it is generally accepted that we must assume things do not exist unless we have evidence that they do. Even theists follow this rule most of the time; they don't believe in unicorns, even though they can't conclusively prove that no unicorns exist anywhere.

To assume that God exists is to make an assumption which probably cannot be tested. We cannot make an exhaustive search of everywhere God might be to prove that he doesn't exist anywhere. So the sceptical atheist assumes by default that God does not exist, since that is an assumption we can test.

Those who profess strong atheism usually do not claim that no sort of God exists; instead, they generally restrict their claims so as to cover varieties of God described by followers of various religions. So whilst it may be impossible to prove conclusively that no God exists, it may be possible to prove that (say) a God as described by a particular religious book does not exist. It may even be possible to prove that no God described by any present-day religion exists.
In practice, believing that no God described by any religion exists is very close to believing that no God exists. However, it is sufficiently different that counter-arguments based on the impossibility of disproving every kind of God are not really applicable.