I can try to explain, but there are people who know much more about this stuff than I do, so hopefully someone more knowledgeable steps in to check my work.
What does ‘random’ or ‘noise’ mean? In this context, random means that any given bit of information is equally as likely to be a 1 or a 0. Noise means a collection of information that is either random or unimportant/non-useful.
So, you say “Compression saves on redundant data”. Well, if we think that through, and consider the definitions I’ve given above, we will reason that ‘random noise’ either doesn’t have redundant information (due to the randomness), or that much of the information is not useful (due to its characteristic as noise).
I think that’s what the person is describing. Does that help?