The three mirrors were the viewers’ Desire, Fear, and Regret.
Lemongrab desired playing with his mother, PB, who in the mirror accepted everything about him and wanted him to be with her. He feared being replaced by Lemonhope, who became every part the ruler he was, down to his tyrant behavior. He regretted the fight over Lemonsweets, and chose to jump into this mirror to stop the fight and save his previous selves, by saving the doll.
Finn desired stepping into CB’s position and dancing with Flame Princess, to be with her again. He feared Jake and BMO eating Finncakes without him, and them being alone. His regret is much less clear than LG’s, though since it is triggered by hearing LG’s voice Finn’s regret may be letting his focus slip off of his mission and allowing himself to be tempted.
In both cases, accepting their regret moved them forward, and ultimately brought them both to meet themselves and Matthew.