To become involved in a relationship.
The implication is that one is caught, or possessed. Usage is questionable -- "get" is never conjugated, because it is usually preceded by another verb.
Often used
tongue-in-cheek or in a pejorative manner, in the case of someone who is actively looking for a partner. Normally, the latter
inference refers only to a sexual relation.
Your ex was at the club last night, man, looking to get got.
I need a girl
-- what's a guy got to do to get got in
this town?