singular/plural noun and agreement

I often revise what I write even as I am writing it.
Do you have a specific question about the sentence you quoted?

Is it correct to use "is" in this sentence?

The subject of the sentence is "It" ("It's" being a contraction of "It is").
"It" is singular.
The word "that" is a relative pronoun which relates to "It" ( not "trees").
"That" is therefore singular and requires the singular verb "is".

I don't know the context, but I would assume that "it" is a TV program or a
magazine article that the author is recommending.

Syntactically, your parsing is correct.
But logically, here I think the writer means to have a Christmas tree in a house is a must for every American family. If you agree with me over this point, then "Christmas trees" and "that is a must ..." sound not quite right, right?