What does to let justice provail at the gate mean?

In ancient times, the wise men would sit at the gate. Townspeople often brought their questions or issues to the men at the gate in search of a justice. In the book of Ruth, chapter 4, you can see this illustrated when Boaz seeks the hand of Ruth but knows that there is another (the near kinsman) with a greater claim. Boaz calls the kinsman in front of the wise men at the gate and convinces him, before these witnesses, to relinquish his claim.

While it may not seem like justice was done by our way of thinking, justice was certainly done in their minds because the actions were consistant with Jewish laws and customs.