The two hosts of “Men in Blazers,” a scrappy soccer-analysis show on the NBC Sports Network, wave handmade signs on the air and broadcast from a room so cramped that guests have to sit behind the duo like children in the back seat of their parents’ car.

But behind all the low-tech trappings, the Men in Blazers are a valuable brand. The TV show and podcast have emerged as a conduit for soccer’s top teams, corporate sponsors and the sport’s growing legion of American fans, much to the exuberance of the show’s England-born,...