Learn Russian: Закончив работу, корректор проверил обособление всех оборотов. - Having finished the work, the proofreader checked the isolation of all clauses.

The phrase Закончив работу is a verbal adverb clause (a деепричастный оборот). In Russian, these clauses must almost always be isolated (set off by commas) from the main clause, regardless of whether they come before, inside, or after the main clause.

How do we know Закончив means "having finished" and not "while finishing"?

Закончив is a perfective verbal adverb, formed from the perfective verb закончить. Perfective verbal adverbs typically end in -в or -вши and describe an action completed before the action of the main verb (проверил). If it were imperfective ("while finishing"), it would be заканчивая.

What exactly does обособление mean here?

It translates to "isolation" or "detachment." In Russian grammar, обособление refers specifically to setting apart a part of a sentence with punctuation (usually commas). This sentence is actually a grammatical meta-joke: the proofreader is checking the punctuation of phrases exactly like the one that starts this sentence!

Does оборотов just mean "turns"?

Literally, оборот means a turn, rotation, or revolution. However, in linguistics, an оборот is a set phrase or a specific syntactic construction (like a clause). For example, the opening clause of this sentence is a деепричастный оборот. Here, it is in the genitive plural (оборотов) because it follows всех (of all).