The beit din's authority descends from the Sanhedrin described in the Mishnah, the rabbinic court system of the Second Temple period that sat in panels, twenty three members for capital cases and seventy one for the highest court in Jerusalem, exercising both religious and, under Roman tolerance, some civil jurisdiction. The destruction of the Temple in the year 70 removed the political centre those courts had depended on, and the rabbis who reconstituted Jewish legal authority afterward, gathering around Yohanan ben Zakkai at Yavneh, did so specifically by relocating the work of judgment away from a single central court and toward a network of qualified judges who could convene wherever Jews lived.
The three judge beit din became the practical, portable unit that resulted: small enough to convene in any community with the required expertise, and sufficient for the matters, conversion, the writing and delivery of a religious divorce, and disputes between litigants who accept its authority, that diaspora communities without the coercive power of a state needed to keep functioning. Its rulings carry weight because the parties involved recognize its legitimacy, not because any government enforces its decisions, which is exactly the arrangement that let Jewish law continue operating for two thousand years without a state to back it.