The noun garter denotes a band worn around the leg to keep a stocking or sock up. The Aberdeen’s Journal (Aberdeenshire, Scotland) of 4^th July 1749 reported the following in its foreign news:

Boston, April 17. Last Tuesday Morning Robert Hunt, who was committed to Goal, for shooting a Lad on the 17^th Instant, (as already mentioned) hanged himself in his Garters, one End of which he tied to one of the Iron Bars in the Window, and was found on his Knees when Mr. Young’s People went in with his Breakfast. The Jury of Enquiry brought in their Verdict Self Murder, and in the Afternoon his Carcase was carried in a Cart to the Veck, and was buried near the Gallows, having a Stake drove through it according to Law.
And Last Thursday Night the poor unfortunate Youth (and only Son) that was wounded by the said Robert Hunt, died.

The phrase to have someone’s guts for garters is used as a hyperbolical threat. It is first recorded in The Scottish historie of Iames the fourth, slaine at Flodden Entermixed with a pleasant comedie, presented by Oboram King of Fayeries, first published in London in 1598, by the English writer and playwright Robert Greene (1558-92)—at that time, disembowelment was an actual punishment:

Ile make garters of thy guttes,
Thou villaine if thou enter this office.

Likewise, in The fountaine of selfe-loue. Or Cynthias reuels (London, 1601), a satire by the English poet and playwright Benjamin ‘Ben’ Jonson (circa 1573-1637), Anaides declares:

Sir, I will garter my hose with your guttes; and that shall be all.

The threats were sometimes carried out; on 18^th December 1790, The Northampton Mercury (Northampton, Northamptonshire) reported the following:

Monday morning, Edward Welsh was executed before Newgate for the murder of Margaret Lane. […]
It appeared on the trial of Lane, that the prisoner and the deceased lived together as man and wife, and lodged in a house in Dyot-street, St. Giles’s. On Saturday last the prisoner came home, but not finding his wife within, he exclaimed, in the hearing of Mary Burke, a witness on the trial that he would be damned but he would have her life that night. Burke begged he would not quarrel with her. He replied again, he would be damned but he would rip her, and give her guts for garters. Soon after the deceased came in, and the prisoner followed her up stairs. In a few minutes she was heard to cry out murder, and on Mary Burke going up, he had just stabbed her in the groin. Her stays and petticoats were immediately cut off, and her intestines were discovered coming out. The prisoner pretended the deceased had wounded herself with her scissars [sic] hanging by her side; but the bloody clasp knife was then laying on the floor. The poor creature was taken to the Middlesex hospital, where she languished till Monday, and then expired.

The striking image, aided by the alliteration in g, have helped to keep the phrase in common usage. An article about Morecambe, Lancashire, published in The Illustrated London News of November 1978, contains:

Everyone is very proud that Eric Morecambe*, the comedian of Morecambe and Wise, takes his name from the town he was born in. […]
There was another comedian, who made the town the butt of his turn, and there was nowt they could do about it, though they’d have had his guts for garters. Colin Crompton was his name, well known in the North Country for dour acts in flat caps. His patter began: “Did you know … they’ve just installed a set of traffic lights in Morecambe. There’s summat to do at night now in Morecambe. If you don’t fancy bingo, or standing up dead in the bus shelter, you can watch the traffic lights change—in Morecambe.”

(* The English comedian Eric Morecambe (John Eric Bartholomew – 1926-84); in 1941, he formed a double act with the English comedian Ernie Wise (Ernest Wiseman – 1925-99) that led to the enduringly popular TV series The Morecambe and Wise Show (1961-76).)

In Idioms as they are spoke, published in The Guardian (London) of 3^rd February 1977, Michael Parkin wrote:

Mr Tony Cowie and Mr Ronald Mackin are busy marking out a clear path for foreigners through the minefield of English idioms by compiling the second and final volume of their Oxford Dictionary of Current Idiomatic English.
Even with the dictionary to hand, a foreigner could still find the occasional English idiom blowing up in his face. For example, Mr Cowie, a senior lecturer in English at Leeds University, was considering yesterday the idiom, “I’ll have your guts for garters.” He thought it should be included in the dictionary with the cautionary note that this was an “informal” expression, defined as being “intimate rather than distant; spoken rather than written; modest rather than grand and imposing.”
It was not an expression strong enough, he thought, to carry the label “taboo,” reserved for idioms “generally avoided by educated male speakers when in the company of women and children . . . best avoided by foreign students.” But what would happen if, say, an Indian student trying to be informal and jocular said to the vicar’s wife: “Aha! Mrs Moore, I’ll have your guts for garters”?