guerre is a war. I should have inferred this from the capitalization of the phrase « Seconde Guerre mondiale ».
lors de la guerre then becomes "lors de la war", or "during the war" - a compound "transitive" preposition. (TIL about the idea of transitive and intransitive prepositions.)
fut is some past tense thing; seems to be the simple past of être (to be).
se déroula is like de-rolling - unrolling, in this use more abstractly as in "to unfold, to proceed", or, as Google translates it, "took place". I can't quite figure out the se part.
d'août is just the month of August!
était is the imperfect past tense, again of être.
près de is what you get when you press things together: they end up close to each other. Specifically, près de 40 000 militaires are nearly 40,000 soldiers.
il s'agit frustrates my attempt to find its idiomatic role. It's clearly the "it is about" meaning of the verb agir - to act, but I can't coax Google to translate "it is" back to s'agit without adding an explicit "about" which clearly doesn't belong in the context of "it is the largest deployment".
depuis is just "since".