It's possible (though generally geometrically more difficult) to prove a negative. The canonical example is, to prove the statement "Fred never lies" you would have to examine his entire life somehow to have any certainty of that -- particularly since it's commonly accepted that virtually everyone lies at least on occasion / a little. In this case, the burden of proof, as well as the required effort, is high. On the other hand ... it's easy to prove that in a specific matter, "Fred lied about x" or even "Fred is a liar" (since you'd only have to demonstrate that Fred lied one time to prove that).

I think when people say this, they just mean that it's not practical for a given question and/or the burden of proof isn't on them because they're not making the positive truth claim in the matter. If someone says "my god exists" it's up to them to demonstrate that this is so -- not up to me to prove that it's not, especially if I simply see no reason to consider that a credible claim, don't consider it likely to be true and don't see any evidence to support it. Of course since most people's gods are supernatural and invisible, they are non-falsifiable, so there is no way to prove OR disprove their existence. It comes down to a question of your belief position rather than any knowledge claims. There IS no knowledge claim for anything supernatural.

What are you ? A lawyer?