The Greek word (ἁγίων) used for "saints" in verse 52 is a form of the same word (ἁγίαν) that is used in verse 53 to mean "holy".
(It's also part of the name "Holy Ghost" or "Holy Spirit".)
Moses's brother Aaron was referred to as "the saint of the LORD" in Psalms 106:16.
Many places in the New Testament refers to Christians as "saints", such as: Acts 9:13 "much evil he hath done to thy saints at Jerusalem", and Acts 9:32 "the saints which dwelt at Lydda".
So in verse 52, "saints" would refer to followers of Jesus (and possibly well respected followers of Judaism) that had recently died.
John 11 describes how, after being buried for several days, Mary and Martha's brother Lazarus was physically resurrected and returned to his family to live out the rest of his new life.
In the same way, the bodies of the saints of verse 52 would have been raised back into physical life.
They would have of course left the graveyards and gone back into Jerusalem to rejoin their families, wondering what had happened.
Those families are the "many" of verse 53.