I don't know what these "specific amounts" comments were, but I suspect the point was to convey that the partitive is (generally) used for undefined amounts, partial things, and uncountable stuff. One kilo is a whole kilo, but the kilo in Finnish in the sentence above is not in the partitive. Lihaa is, but we're not talking about one whole meat, because such a thing doesn't exist. Meat is an uncountable substance, and hence is in the partitive. Whole animals do exist: yksi lehmä, not yksi lehmää.
As a sidenote, even when counting discrete amounts, only one of something is in the nominative, e.g. yksi kilo. Fractions and more than one involve the partitive: puoli kiloa, kaksi kiloa. Or e.g. puoli lehmää if you had half a cow (carcass). This is a somewhat separate usage of the partitive, more of a rule for counting things than purely context-driven from partialness or the lack thereof.
No. You can't just translate word to word or component to component, you need to think about the meanings. The "of" in English has no direct equivalent in the Finnish sentence, it's just implied that the weight measure given is about the meat, due to them appearing in the same sentence/clause.