To make a full disclosure; to confess.
'Come clean' originated in the USA in either the late 19th or early 20th century. The earliest example that I have found of it in print is from the Moberly Evening Democrat, August 1904:
"Now, then Chillicothe papers - Constitution, Democrat, Tribune - come clean. Tell the truth."
The unnamed journalist who wrote that must have liked the phrase, as it appears several times in the same paper from around that time and before I can find it elsewhere. It's possible that the phrase was coined in Moberly, but that's just conjecture.