Why "jets" is written with an apostrophe in this sentence: Russian attack jets' close call with US warship riskiest encounter in years

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The apostrophe indicates possession (like "the man's book"). When the subject (that is possessing an object) ends in "s" then the apostrophe can be put on the end (after the "s") to indicate possession instead of doing it like "jets's" (which looks sort of unnatural).

So it's saying the "close call" belongs to the "Russian attack jets" — The "Russian attack jets" had a "close call" with a US warship.

A "close call" can refer to a near miss (e.g. almost hit into them) or a narrow escape from danger (e.g. just managed to avoid starting an accidental battle/conflict).

"Riskiest encounter in years" in this context means that this was the riskiest or most dangerous encounter the Russian Jets had with a US warship in a long time.

Note: the grammar in this sentence is only acceptable in "Headline English" for reporting news headlines. It is not acceptable in regular written English (like normal paragraphs).