Not sure if I can explain it in text ... but I'll take a stab at it.

When an airplane banks and turns, the lift component is no longer perpindicular to the ground. There becomes a horizontal component of the lift. This introduces a sideslip in the turn.

For an aircraft that has a dihedral wing (that is a V design as apposed to a ^ design), during a turn, the wing that is lowered in the turn has a higher angle of attack due to the sideslip which I mentioned above. Because the lower wing has a higher angle of attack, it creates a rolling moment back in the opposite direction (a roll opposite of the banked turn). Therefore, this creates stability in the roll axis.