To “spill the beans” means to reveal a secret. “Spill the beans” has been cited in print since at least 1899, when it was cited in horse racing, meaning something similar to “upset (of the apple cart).” A pot full of beans is almost certain to spill as it is cooked and carried; only when some beans have been spilled have beans truly been served.
 
Eventually, to “spill the beans” would mean to reveal secrets or information.
 
“I used to be a member of the secret cooking society. They kicked me out for spilling the beans” is a food pun on the term.
 
“Spill the tea” (to tell the truth, with “tea” or “T” meaning “truth”) is a related expression.
     
     
Wiktionary: spill the beans
Verb
to spill the beans (third-person singular simple present spills the beans, present participle spilling the beans, simple past and past participle spilled the beans)
1.(idiomatic, intransitive) To reveal a secret; to disclose.
 
Wikipedia: Don’t spill the beans (idiom)
Don’t spill the beans is an English language idiom That means “to inopportunely or accidentally reveal a secret.”
 
There are a number of folk etymology hypotheses for its origin, some attributing it to the ancient Greek voting methodology of gathering beans in a jar, or to agricultural origins. However, the phrase first came into use in the early 20th century, in America, which tends to not fit in with these explanations. It is likely that the phrase originated within the American Great Depression, among the transient poor under-culture (the hobo culture).
 
The Phrase Finder
Spill the beans
(...)
The earliest uses of ‘spill the beans’ come from the USA. The meaning of the phrase was then something like ‘spoil the beans’ or ‘upset the applecart’, which harks back to the supposed Greek knocking over of a bean container. The first example I can find is from The Stevens Point Journal, June 1908:
(...)
We have ‘spill’, meaning ‘divulge’, but why beans? Well, it could have been almost anything. In fact, there are several ‘spill the’ variants - ‘spill the soup’, ‘spill your guts’, or simply, just ‘spill’.
 
(Oxford English Dictionary)
to spill the beans: to reveal a secret. slang (orig. U.S.).
1919 T. K. Holmes Man from Tall Timber xxviii. 355   ‘Mother certainly has spilled the beans!’ thought Stafford in vast amusement.
1921 R. D. Paine Comrades of Rolling Ocean viii. 136   The beans are spilled, and that is what Maddigan guessed the moment he set eyes on you.
1928 Daily Express 10 Oct. 6   Spill the beans. Come clean on the whole game.
     
11 February 1899, St. Louis (MO) Post-Dispatch, “The California Oaks,” pg. 5, col. 3:
Trexler certainly did spill the beans for the little Texan, whether intentionally or not.
 
14 November 1901, The Enquirer (Cincinnati, OH), “Latonia Gossip,” pg. 4, col. 6:
“The fact is the Schnell-Laufer people were afraid of Jesee Jarboe and they gave Mose Goldblatt $100 to scratch her. Then they went in for a plunge, bet the money all over, and what does Nobleman do but come along and spill the beans.”
 
21 April 1902, The Enquirer (Cincinnati, OH), “Coup: Intended With a Horse Which Was Given a Moonlight Trial at the Latonia Track,” pg. 3, col. 7:
Names are withheld for the reason that it would be regarded as exceedingly rude to spill the beans on the eve of the feast, but there is no impropriety in discussing the plans.
 
Chronicling America
25 November 1902, St. Louis (MO) Republic, “Maginn Looks Ahead to $50,000 Handicap,” pg. 6, col. 3:
“Ethylene was 15 to 1 one day and would have won sure had Battiste been up. He was set down by the starter in St. Louis and I had to ride a bad boy. Then we put Battiste up later and got down. Kiley told him to take her back a bit. He did, and in some manner the field ran around and over her so that she was shut in, cut off and lost. So the beans were spilled.”
 
31 March 1905, Cleveland (OH) Leader, pg. 8, col. 2:
ST. JOE BEATS COMISKEY’S SQUAD.
ST. JOSEPH, Mo., March 30.—(Special.)—Comiskey’s squad of the White Stocking aggregation were the first to spill the beans by being defeated to-day by the Saints by the score of 2 to 3.
 
26 March 1908, New Orleans (LA) Item, pg. 2, col. 5:
John McBride, long overdue, was the one to spill the beans in the closing event.
 
1 June 1908, Atlantic (IA) Evening News, “Former Blacksmith a Power in Congress,” pg. 7, col. 2:
Tawney, when he came to congress, wasn’t welcomed within the big tent. He had to wait around on the outside. Then the blacksmith (Jim Tawney—ed.) got busy. He just walked off the reservation, taking enough insurgent Republicans with him to spill the beans for the big five.
 
27 August 1908, Rockford (IL) Republic, “Another for the Cubs,” pg. 3, col. 3:
The Cubs just sat still in the boat, laughing up their sleeves and waiting for the runs to be forced upon them. They had nothing to worry them, confidently expecting the tailenders to spill the beans at any minute, which they did in grand style.
 
Chronicling America
10 January 1910, Los Angeles (CA) Herald, “M’Cary should take mortgage on Langford’s life right away” by Jay Davidson, pg. 6, col. 1:
Flynn May Spill Beans
Of course there is a possibility that Jimmy Flynn may spill the beans for Sammy, and then another scrap between them would be in order.
 
9 October 1915, New Castle (PA) News, pg. 4, col. 2:
“How big was Alexander, pa, that people called him great?” “Well, my son, he was big enough to spill the beans, whatever that means.”
 
OCLC WorldCat record
Let’s spill the beans : fox trot
Author: Harry Revel; Mack Gordon; Jack Mason
Publisher: New York, N.Y. : Crawford Music, ©1935.
 
OCLC WorldCat record
Don’t spill the beans
Author: Hasbro, Inc.
Publisher: Pawtucket, R.I. : Hasbro, ©1999.
Edition/Format:  Game : Game Visual material : English
Summary: Drop the beans into the pot as it wobbles to and fro, but don’t spill the beans or the game is over! For two to four players.
Edition/Format:  Musical score : Motion picture music : English
Publication: WFAA collection.