While I'll admit I haven't watched the video linked to, I think what might be being referred to here is what's called "Equivalence". It has more to do with the comment about losing light (DX vs FX) than sharpness, however. Maybe I should watch the video because I associate equivalence with "noise" more than sharpness... *scratches head*
Anyway, you may want to see this article:
What is Equivalence and Why Should I Care?
Pertinent Quote from article:
You are no longer capturing all the light that the lens is projecting [with a DX lens]. The light intensity on the sensor remains the same, but the amount of light you can capture has dropped. (Because you're now only capturing the inner, bronze colored cone of light, rather than the more yellow one).