The pervasive threat of jamming attacks, particularly from adaptive jammers capable of optimizing their strategies, poses a significant challenge to the security and reliability of wireless communications. This paper addresses this issue by investigating anti-jamming communications empowered by an active reconfigurable intelligent surface. The strategic interaction between the legitimate system and the adaptive jammer is modeled as a Stackelberg game, where the legitimate user, acting as the leader, proactively designs its strategy while anticipating the jammer's optimal response. We prove the existence of the Stackelberg equilibrium and derive it using a backward induction method. Particularly, the jammer's optimal strategy is embedded into the leader's problem, resulting in a bi-level optimization that jointly considers legitimate transmit power, transmit/receive beamformers, and active reflection. We tackle this complex, non-convex problem by using a block coordinate descent framework, wherein subproblems are iteratively solved via convex relaxation and successive convex approximation techniques. Simulation results demonstrate the significant superiority of the proposed active RIS-assisted scheme in enhancing legitimate transmissions and degrading jamming effects compared to baseline schemes across various scenarios. These findings highlight the effectiveness of combining active RIS technology with a strategic game-theoretic framework for anti-jamming communications.