I'm gonna use my wings scenario here; it's my favorite.
My claim: "I have retractable wings."
Naturally, knowing human anatomy, we have more than enough reason to be dubious of this claim right? Well after you ask me for proof and my response was "It's not my job to prove it true, you have to prove that I don't."
Do you see how this logically doesn't follow? Why is it up to you to prove a negative? Can you prove I don't have wings? No. Here's why.
You look behind me. No wings. Caught me?
They're under my jacket.
You feel my back for wings. No wings. Caught me?
My wings are small.
You feel a more fine area for smaller wings. No wings. Caught me?
My wings are very thin.
You get frustrated and yank my jacket off. No wings. Caught me?
Nope. My wings are under my skin.
See? No matter what you do. I can create a new scenario with which my wings exist and can keep doing so, no matter how ludicrous it gets. My claim that I have the wings will still stand even though I have not proved it. And all you've proven and continue to prove is that under a,b,c scenario I do not have wings, but what about d,e,f?
This craziness is easily rectified by withholding belief in my claim until it is proven true. Until I show you myself that I indeed have wings, you're justified in calling my claim false. Because I have not proven it to be true in any scenario, It is not true that my wings reside under my jacket, under my skin. And you're justified to not believe so until I prove to you that the claim is true.
It's the same with people who ask atheists to prove God does not exist. We can't do that. The scenarios play out similarly to the wings. Scenario A, Scenario B, Scenario C, Scenario D, to the infinite. If we tried to prove a negative your claim would remain true forever.
But because that's not how logic works, we withhold our belief until the claim is proven true. And are under no contract to prove it false. Burden of proof.
If I didn't explain it well enough (as I do tend to be scramble minded at 4:43 AM sorry) here's an excellent example of what I just explained from James Randi.