Cry off

Can anyone explain the origins of the phrase 'to cry off'?

Since the 18th century it has been used much as the modern synonyms are used, the modern synonyms being to beg off or to call off, the latter having a slightly different sphere of usages. The definition in the Oxford English Dictionary explains to a degree how cry + off came to mean what they do.

"20. cry off. intr. To exclaim that a negotiation is broken off, on the part of the exclaimer; to announce one's withdrawal from a negotiation, treaty, engagement, etc.

"1775 SHERIDAN Rivals III. i, I should never be the man to bid you cry off. 1857 TROLLOPE Three Clerks xviii, Would she be the first to cry off from such a bargain? 1890 G. M. FENN Double Knot I. Prol. iv. 62 He soon cried off on finding that his challenge was taken up."

This definition explains the use of the verb "cry," but the meaning nowadays isn't so much dependent on the crying out as the fact of breaking off a negotiation or breaking a promise.
SS

The original meaning of "cry" was 'entreat, beg, beseech, implore, in a loud and moved or excited voice' (it derives from Latin "gridare"). For centuries one of its routine meanings was 'announce publicly so as to be heard by all concerned; to give oral public notice of, to proclaim; to appoint or ordain by proclamation'. You could 'cry' an event such as a meeting, or a public entertainment: so logically, if you needed to inform people that the event was cancelled, you "cried it off". (VSD)