Mario can wear this badge for one BP, and in battle by paying one FP, he can increase his attack power for his next attack. This effect is stack-able with other turns he uses "Charge."
Charge is found near the end of Chapter two. In the bottom portions of The Great Tree. Mario can climb from pillar to pillar and eventually get onto the same spot as the pillar that has this badge on it. Although, he must fill the room with water first.