The wall molecules closest to the inside are the warmest, which means they' re bouncing vigorously about because heat makes everything move faster. Molecules farther from the inside warmth are more lethargic. They are hunkered down against the increased cold. At the boundary where warm molecules bump into cold molecules, conflict arises. Warm molecules bang into cold molecules and in the collisions, they give up some of their kinetic energy to their cooler kin.
Conduction moves heat from hot to cold.
The transfer is strictly one-way. Heat never moves from cold to hot. An ice cube, for example, chills a drink because its cooler molecules absorb warmth from the surrounding liquid. It never makes the drink warmer. The second process, convection, works more circuitously.