Origin and meaning of phrase "cut the cheese"

Cut the cheese:

• (Slang) to release intestinal gas. (Crude. Use caution with the topic.) - Who cut the cheese? People who cut the mustard in the car have to get out and walk.

(TFD)

It’s difficult to track down the origin of this expression, but according the The Phrase Finder:

• “Cut” has been used for flatulence since the 1800s, as testified by several sources and continued today in the mainly American expression, “To cut a fart”.

• Rude Boy says cheese was introduced to the mix in the late 1960s, citing the Dictionary of American Regional English.

• While according to The Concise New Partridge Dictionary of Slang and Unconventional English the expression is an AmE one from 1959.

The following example is from The Definitive Fart Book - 1961:

• Funny, everybody does it, but nobody wants anybody to know they're the somebody who: Cut the Cheese, Passed the Gas, Let One Rip, Shot a Bunny, Copped a Pop, Popped a Bubble, Cranked a Smoker, Pinched an Egg, Split the ...

Probably the simple and more intuitive answer is the right one, referring to the strong odour that emanates when the rind is cut on some of the more pungent cheese varities.