If it were "this much", you'd need to say ĉi tiom instead of just tiom. This is similar how tiu means "that" and tiu ĉi means "this". Similarly, tie means "there" and tie ĉi means "here", etc. The words that start with ti- always indicate something removed from the speaker (distal), and you need ĉi to make them mean something close at hand (proximal).
That is certainly the case within the Duolingo course - which requires a ĉi for something to be translated proximally ("this" or "here"). Outside of the course, it's a little messier. When there is only one thing, there is often no difference between "this" and "that" - and there's no reason to include ĉi in those circumstances.
So, with that in mind "you eat this much meat" is not really wrong here, but the course will mark it wrong, and since we know that in advance, we can give the course what it wants.
No, it means "this much" or "so much". For "a lot", you'd say multe da, as in Vi manĝas multe da viando, meaning "You eat a lot of meat."