The phrase я люблю есть горячим means "I like eating while my body temperature is high" (literally, "I like eating while I'm hot").
Note that in Russian, unlike English, the phrase я горячий doesn't mean "I'm sexy" or "I'm feeling hot". It only means what it says on the tin, like "I'm tall" or "I'm black" do. Even then, it's not idiomatically used to say "I have a fever" — not in the first person. The phrase я люблю есть горячим makes little sense to a Russian speaker.
I don't speak French, but it looks to me like it's being used here as a pedagogical device.
In Russian, you can put the modifier of a direct object of a verb in the instrumental to convey a meaning of "while":
- Живыми брать демонов! // The demons are to be taken (while) alive!
- Полюби меня чёрненьким, а беленьким меня всякий полюбит // Love me (while I'm) black ("at rock bottom"), for anyone will love me (while I'm) white ("at top").
- Месть — это блюдо, которое подают холодным // Revenge is a dish best served (while) cold
This only works when the verb has a direct object modified by the adjective in the instrumental.
If there's no such object (i.e. when the adjective is used as a collective noun), you should put it into the singular neuter nominative: ешь свежее "eat fresh", покупай местное "buy local" etc.
This answer probably wants to convey something like "for the purposes of learning French, imagine that the adjective in question is not a collective noun, but a modifier of an omitted direct object".
But again, I don't speak any French and this is just my theory.