Short answer: It was a mistranslation of a phrase from the Vulgate Bible.
Giving someone the “cold shoulder” is a phrase commonly used to express the dismissing or ignoring of someone.
The most common origin of the phrase is actually folk etymology. It goes something like this:
In the early 1900’s it was customary for a hostess to serve hot meat to welcomed customers and cold meat to those who had overstayed their welcome. Since the cold meat given to unwanted customers was usually mutton, the hostess would say, “give him the cold shoulder” –of mutton. This is an accurate example of how the phrase was being used in the early 1900’s, but it’s not the original origin.
Wikipedia summarizes the actual origin nicely:
The first recorded use of the expression was in 1816 by Sir Walter Scott in the Scots language, in The Antiquary. This expression is a mistranslation of the Latin phrase “dederunt umerum recedentem” from the Book of Nehemiah 9.29 from the Vulgate Bible, which actually means “stubbornly they turned their backs on you”, which comes from the Septuagint Bible’s Greek equivalent ἔδωκαν νῶτον ἀπειθοῦντα. Latin umerus (often misspelled humerus) means both “shoulder” and “back”:
“Ye may mind that the Countess’s dislike did na gang farther at first than just shewing o’ the cauld shouther…”
where “cauld” is the equivalent of cold and “shouther” means shoulder, which is further supported by contextual usage
in The Antiquary. Neither eating nor food is expressed in the passage, but the phrase is presented in a rather allusive way. The phrase also appears in one of Scott’s later works, St. Ronan’s Well and after the 1820s it had traveled to America. Dated June 1839 in a letter to the editor in the New England newspaper The Bangor Daily Whig and Courier:
“… eminent individuals and his cabinet advisers turned “the cold shoulder” to their ambassador, for his independent act upon this occasion.”