to stay behind: to remain in a place after others have gone on
The word is built from two smaller words meaning "under" and "remain," so at its most concrete it simply means to stay put while others move on. In the boyhood story of Jesus, the boy stays behind in Jerusalem after his parents have set out for home (ὑπέμεινεν in Luke 2:43), and in Acts, Silas and Timothy stay behind in Beroea while Paul travels ahead (Acts 17:14).
This plain sense of holding your place is the root underneath the word's more famous use. To endure, in the next meaning, is really just to hold your place under pressure rather than under travel.