As a native speaker of American English, I would only ever say "back home" and never insert a preposition between the two. I would say, "I'm going back home" when someone asks, for example, how I will spend my vacation. In this case, "home" refers to where I was raised. If I just say "I'm going home", this could just mean the place I live now OR the place I was raised.
You can use "at" with "home" as in, "I am spending the night at home" or "Cooking at home is less expensive than eating out."
If you want to use "back to" in the above context, you might tack "my house" on the end of it.
Q: Where to you want to go after dinner?
A: Let's go back to my house.
You can also use at in this context:
Q: Where did you say the blueprints were?
A: I left them back at my house OR I left them back at the house (implies
everyone knows which house is being
spoken about, but you do not
necessarily own it.
I cannot speak for how the above usage may differ in British English.
Both are correct. There are instances where they mean the same thing and some instances where they don't.
If you were telling someone that you didn't intend to go out tonight, you could use either.
"Do you want to go with us to a restaurant tonight?"
"No, I think I'll stay home."
"No, I think I'll stay at home."
However, if someone were asking where you were staying, where the answer might be "a hotel" or "a friend's house", you would definitely say "at home".
"Are you going to get a hotel room for the conference?"
"No, I'm going to stay at home."
"No, I'm going to stay home."
(Note that if you did say the struck-through version, that would tend to imply that you weren't going to go to the conference at all.)
However, that "home" is uncommon in that it also functions as an adverb. You can never leave the preposition out with other similar nouns. For example, this is clearly wrong:
"I'll probably stay work."
In that case, you have to create an adverbial prepositional phrase:
"I'll probably stay at work."
Best Answer
Both are widely used and understood. There's even a third alternative, "home of". None of the three is "wrong" in and of itself. However, they mean slightly different things, and in your particular context, "to" is the preposition of choice.
"To be home to X" is a set phrase used precisely for talking about inhabitants of a given place (be it plants, animals, people, or robots). In your sentence in particular, it implies that the diverse flora is already there and has been for quite some time, while home for would imply that the diverse flora is kind of wandering around looking for a home, and now we can offer it one at the Himalayas.
Just to round it up, here are some quick-and-dirty Google stats:
And here are the stats from the Corpus of Contemporary American English (COCA) and the British National Corpus (BNC):