From a stage in front of rows of picnic tables under bright blue tents, Dennis Hohloch tells the crowd that his country doesn’t need more watchdogs to peg Alternative for Germany as far-right extremists that oppose the constitution. He says the party will renew democracy and freedom. The crowd of more than 100 people erupts in applause for Mr. Hohloch, a member of the AfD federal executive board and the parliamentary secretary for the group in Brandenburg. While this could be mistaken for a political rally, it is actually the AfD’s summer festival in Senftenberg, a city of about 23,000...