It's just because the possessive needs a preposition here, and the implicit one in 'hjemme' doesn't do the trick ('hjemme' is actually a dative form, so a special case). Think of it as "They are (at) home at mine", but in the sense of actually being present at your home, not feeling at home.
If we wanted to write this sentence the "English" way, we'd say:
De er i hjemmet mitt
De er i mitt hjem
They are in/at my home
But, while being a valid construction, it sounds strange in this setting.
That meaning can sometimes be implied, but not assumed. 'hjemme med meg' would be a clearer way of expressing it.
The sentence we're discussing means that 'they' (referring to either living beings or inanimate objects) are at my home, though in most situations I could see myself uttering it, I myself would be elsewhere.
Thank you! Yes, we would use "of " instead of "at" in English for this meaning. Prepositions are often different from language to language. " at the home of mine" is so long though that we would just use "at my home", but perhaps it will help someone to understand. I really appreciate all the time you are spending explaining this all to us.
"Hos" doesn't have an easy, direct translation to English, but works the same as the French "chez", if you're at all familiar with that?
I would suggest thinking about "hos" as meaning something like 'in the home of' or 'at the place belonging to'... So in this sentence if we just said "De er hjemme", I'd assume 'they are at home' in what is most likely their own home. Stating "hjemme hos meg" clarifies that the people are at my home, not their own home or someone else's unspecified home.
Another use can be if I asked you, hey, where's your sister tonight? And you could reply, "Hos Camilla", meaning she's at Camilla's place/house/home.
I don't know if this helps explain it at all? Sorry if not!
Does "De er hos meg" mean something different from "De er hjemme hos meg"? In what context would you use one over the other?
If I had to guess, the former would just mean, "they're at my place" whereas the latter might mean, "they're at home (at home as in 'comfortable')/staying at my place."