The term refers to a pre-marriage ceremony in Colonial New England between a man and a widow; not all such marriages, apparently, but only when the widow had received substantial property from her deceased husband. The custom was that she would appear before witnesses dressed only in her shift (nightgown). The husband would then affirm before the witnesses that he accepted her as she was—in other words, only herself, not the property she had received from her first husband—and renounced any claim to her property, and also made it known that he would not pay off any debts that the widow inherited.
An example, from the Vital Records of Barnstable, Massachusetts:
Nathan Noyes & Mehitable Bangs wid: Sept 20 1750 ye Bride Setting
in ye Bed ye Bridegroom Declared before ye witnesses present that
he took her in her Shift as She was, having never Received
anything of her former Husbands Estate nor expecting to Receive
Any and that he would not pay any of ye debts of it - This
Returned on ye Back of ye Certificate Pr Mr Green Attest David
Crocker Town Clerk.
Apparently variations of the ceremony existed. Other versions have the bride "appearing" naked behind a curtain held by her (hopefully female) friends, or taking place on public property such as a roadway or common area rather than in the bride's or groom's home.