I'm not sure this rigorous (or if this is the proof they were referring to), but...

I'm not sure this rigorous (or if this is the proof they were referring to), but they mentioned t a pretty compelling argument in passing; if you could compress all inputs by at least one bit, then you could use that algorithm again on all the outputs and reduce them by at least one bit, until eventually you've reduced every possible input to one bit, which is clearly not possible (at least, not if you want to ever be able to get back the original value). This means that every compression algorithm either a) can't compress some inputs to a smaller size than they already are or b) lose some of the data in the process. (Lossy compressions can still be useful, of course; that being said, I'd imagine assume that even most lossy compression algorithms to be idempotent).

EDIT: The sibling comment from rmidthun is much more rigorous and concise, so if you're looking for a more proper explanation, ignore this and read that