Going into the 1942 game, BYU hadn’t beaten Utah since 1898. One of their alumni was serving in ww2, and he sent a telegram to the team telling them to “Give ‘em hell, Brigham”.
BYU ended up winning that game, so it’s been a part of their folklore/ battle cry ever since. (They didn’t win another one until 1958, btw)
It’s nothing more than a bunch of adults that have been treated like children their whole lives so they feel like a 8 year old with their friends, swearing, hoping their mothers don’t catch them.