I read this sentence:
In the Arctic tundra, temperatures are below freezing for nine months out of the year.
Why didn't the writer use "during the year"? What is the difference between "during the year" and "out of the year"?
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It's just a style choice. There are always many ways of saying things like this in English and in all other languages. How one decides to say it is usually dependent on context, audience, and the aesthetic judgment of the writer.
I think this is a bit verbose and would be better as:
In the Arctic tundra, temperatures are below freezing nine months of the year.
Some native Anglophones will prefer:
In the Arctic tundra, temperatures are below freezing for nine months of the year.
Still others will say it another way.
The sentence is idiomatic English and not at all grammatically wrong.