When theists claim that atheists believe humanity came from "nothing" are they really saying...?

No, they don't say we came from nothing. They say we came from unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident after unlikely accident. For example, the Big Bang was a cosmic accident in which some unknown source caused the unverse to spark to life (hmm... sounds like Genesis to me if you replace unknown with Godly) then we accidentally got a planet, then neucleotide bases were acidentally created, then they were accidentally put in the right order, then they were accidentally next to something that accidentally synthesized them into proteins, that accidentally created a unicellular organism, that encountered many accidents in it's DNA and ultimately arrived to be every living thing you see today.