We investigate the conditions of the complete tunneling of light across a composite barrier made of multiple layers involving metamaterials. It is shown that complete tunneling phenomena are related to the resonance transmission properties of local modes formed in barrier layers and that there are two distinctive kinds of local barrier modes involved in actual complete tunneling: the degenerate inner-barrier mode and the full barrier mode. Complete tunneling occurs via two successive mode couplings: from the incident plane wave to the plane wave in the transmission layer through the direct mediation of these two kinds of local barrier modes.