1
general::
Confess, make a full admission, as in Come on, Tim, you'd better own up that you lost the car keys. This idiom uses the verb own in the sense of "acknowledge." [Colloquial; mid-1800s]
American Heritage Idioms
2
general::
Phrase(s): own up (to something)
to admit something; to confess to something. • I know you broke the window. Come on and own up to it.
McGrawhill's American Idioms And Phrasal Verbs