The First Crusade began because of the Pope's response to the Byzantine Emperor's request for assistance against the Seljuk Turks. Why did the crusaders' objective transform from helping the Byzantines to capturing Jerusalem along the way?
It's actually not entirely clear why! It's still debated by historians of the crusades. u/J-Force recently gave a lengthy breakdown of why the First Crusade happened the way it did, which will hopefully help answer your question.