During the time period of the film, the wizarding world has laws that make it the head of a household's decision who the other members marry. The younger members have no legal say in who they marry (although they can be granted a divorce in cases of abuse or infidelity).
Newt has been married at least twice before, each time to someone who treated him terribly. Just before the movie, Theseus became head of the household. Theseus genuinely cares for his little brother and just wants him to be with someone who treats him well. Because of this, he decided to ask his penpal, Graves, to marry his brother, because he knows he's a decent guy. Graves, feeling sorry for Newt and wanting to help out his friend, had agreed. However, by this point, Newt's too traumatized and wants nothing to do with arranged marriages anymore (maybe not even relationships in general).
He sets off running from the engagement, traveling from place to place because if he stays in one country too long, the wizarding government might force him to go back to England and get married.
The events of the movie happen, and Graves is rescued. MACUSA keeps Newt from leaving at the end, with the intention of forcing him to go through with the marriage. Graves, however, demands that they let Newt go, because he agreed to the engagement to help Newt, not to traumatize him further, and now that Newt helped save Graves and all of New York, the least they can do is let him come and go as he pleases.
Newt leaves, but grateful for Graves's gesture, starts a correspondence with his fiancée, and they become friends, maybe slowly evolving into something more.