"tamten" is for masculine mouns, "tamto" is for neuter nouns. And then you have "tamta" for feminine.
'mięso' here is the nominative form right? I'm confused, I thought eat and drink took accusative or is that wrong?
You're right, "jeść" (to eat) and "pić" (to drink) take Accusative.
But several genders have the Accusative and Nominative forms identical. So "mięso" here is Accusative, it just looks the same.
Those genders are: neuter singular (like "mięso"), masculine singular inanimate (that's the only situation where it matters whether a noun is animate or not) and 'not masculine-personal plural'.