Whew! Too hard for me!
That cyclic diester of boric acid...that's some reagent that new to me. However, seeing as I took Orgo in 1978, that's not unusual.
My guess is that it's reacting with that allyl halide by adding to the double bond, leaving the bromine on there. Then, the product halide reacts with the triphenylphosphine derivative, makes a Wittig reagent, then that attacks some carbonyl moiety generated on one of those R groups attached to the boron...generated by some rearrangement that's beyond my knowledge.
It's one of those elegant jobbers that makes you say "of course" when somebody shows you the answer and draws all those little 'which electron goes where' arrows.