It does not attack offensively, but rather attempts to steal gil from the party, as its name implies. Once it successfully steals, it escapes on its next turn, taking the gil with it. As with most enemies in Mt. Zozo they have high Evasion, so (if playing the GBA port; see below) magical attacks are advised, particularly Fira. Dispatching them before they can flee will earn back any gil they stole at the end of the battle.
Because of the Evade bug in all versions of Final Fantasy VI apart from the GBA port, Mugbears were much easier to dispatch than intended.