Why do I have to use "את" in this case? Because their food isn't the subject?
If I squint hard and tilt my head a little, it can maybe look like "I like their eater", in the singular, so maybe, the person who is eating them or the person who is eating and belongs to them somehow. But seriously, I need to make an effort to get that. אוכל as a noun is immediately interpreted as "food". :-)
No. That would be אני אוהב לאכול אתהם... "to eat" is an infinitive. You see infinitives with (ל) in front of the verb.
It's needed before a direct object. "The item", proper nouns/names (excluding.immediate family) it's in the discussions if you search direct object & also in the tips and notes.