The phrase the proof of the pudding is in the eating means the quality, efficacy, etc., of something can only be shown by putting it to its intended use. (Here, the word proof means test, i.e. procedure designed to establish the quality, efficacy, etc., of something.)

The notion conveyed by the phrase was evoked in Kyng Alisaunder, a romance composed around 1400 (tasting has the general sense trying, testing):

(1810 edition)
Hit is y-writein, Every thyng,
Himseolf schewith in tastyng.
So hit is of lewed and clerk:
Hit schewith in his werk.
     in contemporary English:
It is written, everything
Itself shows in the tasting.
So it is of either an unlettered or lettered man:
It shows in his work.

The phrase is first recorded in the list of proverbs of Remaines, concerning Britaine: but especially England, and the Inhabitants thereof. Their Languages. Names. Surnames. Allusions. Anagrammes. Armories. Monies. Empreses. Apparell. Artillary. Wise Speeches. Prouerbs. Poesies. Epitaphs (3^rd edition – London, 1623*), by the English historian and herald William Camden (1551-1623):

All the proofe of a pudding, is in the eating.

(* The Oxford English Dictionary (3^rd edition – 2007) erroneously indicates that the phrase appeared in the first edition (1605) of Camden’s work; I have not had access to the second edition (1614).)

Elliptically, the proof of the pudding denotes that which puts something to the test or, in later use, proves a fact or statement (proof being in the latter case understood in its sense of evidence, argument establishing a fact or the truth of something). In Democrats Are Fortunate To Have So Many Orators, published in The Philadelphia Inquirer (Philadelphia, Pennsylvania) on 13^th June 1941, John M. Cummings, reporting from Harrisburg, made fun of a meaningless use of the shortened phrase:

The Democratic State Committee met here yesterday to indorse candidates for Justice of the Supreme Court and Judge of the Superior Court. […]
In Meredith Meyers they have a chairman who belongs to the school of orators who believe that words were intented to convey ideas, not to befuddle or confuse the mind. […]
He used the metaphor to sum up the record of the Democratic House and what he said is worth recording, not merely for this generation, but for posterity.
In this one sizzling summing-up sentence Mr. Meyers said as follows to wit:
“The proof of the pudding is the best judge of the result attained.”
[…]
In our experience in this trade we never heard the achievements of any political party summed up so well in one short sentence. On the way back to the hotel we kept repeating it—“the proof of the pudding is the best judge of the result attained.” In the bar we sprung it on a gentleman deeply absorbed in his beer. He touched his head with his finger and winked at the barkeep. This is a way some people have of indicating their belief there’s a screwball in the joint.
We tried it on six other citizens in various stages of sobriety. And we overheard one citizen say unto another “that Inquirer guy used to be an all right sort of person but he seems a bit tetched lately and how he holds on to the job is a mystery.”
In front of the hotel we walked up to a traffic cop. “Officer,” said your correspondent, calmly and with becoming dignity, “the proof of the pudding is the best judge of the result attained.”
“Yes, sir,” said he. “Go down this street three blocks, turn to the right and in the third house on the left you’ll find one of the best doctors in town.”

illustration for Democrats Are Fortunate To Have So Many Orators
The Philadelphia Inquirer – 13^th June 1941

The variant the proof is in the pudding makes no sense, as the American author, columnist and journalist William Safire (1929-2009) explained in the Daily Press (Newport News, Virginia) on 5^th January 1986:

Assistant Attorney General Stephen Trott told a “Meet the Press” panel that he was pleased with Israeli promises to cooperate in the investigation of espionage in the United States, but added: “We will take them at their word, but the proof is in the pudding.”
The proof is not in the pudding. You could stir around the pudding for hours and never find the proof. The proverb goes: the proof of the pudding is in the eating.
[…]
The point of the proverb is that a pudding may look good, or smell good, or even feel and sound good, but all that is irrelevant and sometimes misleading; the only true test of its success as a pudding is the satisfaction it provides when it is eaten.
The clipping of the proverb to “the proof is in the pudding,” stated as flatly as “the saw is in the cake,” renders the adage meaningless. I would not ordinarily take a pop at a speaker for making this common error, but I expect the head of the Justice Department’s criminal division to be careful about the best-known proverb about proof.