The common English phrase, out with the old, in with the new, means “it is good to replace old things with new things”—“to get rid of the old and welcome the new.” People often say this phrase when getting a new leader, a new job, a new relationship, or a new style, for example.
In this phrase, out means away or gone, just like we say that old fashions are out of fashion and unpopular things are out. And we say that fashionable things are in.