The following is an old article, which should be a very popular explanation of this algorithm:
This algorithm is proposed to solve the "stable matching problem". The so-called stable matching problem refers to: there are n men and n women. Men have their own list of lovers, from their favorite goddess to their least favorite dinosaur, while women have the same list, from their favorite Gao Fushuai to their least favorite wretched man.
Obviously, a woman that a man likes may despise him at all. And men who are liked by many women cannot marry these women at the same time. So it is difficult to find a solution that makes everyone get married and everyone is satisfied.
In order to solve this problem, mathematicians Shapley and Gale put forward a special algorithm in 1962, which was later called Gale &: Shapley algorithm. They proved that according to this algorithm, a stable combination that satisfies everyone can be found, provided that the number of men and women is equal.