The use of backward induction was developed by Selten (1965) is his definition of a subgame-perfect equilibrium in extensive form games. The logical intuition is that a game may be solved by looking at the optimal action of the last mover in the game, assuming that his optimal action is the action taken and then progressing to the next to last period. In the next to last period the optimal action is determined assuming that if the game proceeds to last period the last mover will take her optimal action. This logical structure proceeds back to the beginning of the game at which point the first mover's optimal action is determined given that all following actions are best responses.

To look at an example of backward induction using a common game called the centipede gameclick here.