It's typically used in comedy shows to show a character trying to act smart but failing, or as a sort of short-hand way to show someone is going to give a long-winded, impersonal, and/or cliché speech.

The 1959 crime-drama film The FBI Story uses it in the film's opening sequence unironically, but the joke-y version of it may very well have started with The Simpsons.

In the 1992 The Simpsons episode "Brother, Can You Spare Two Dimes?", the character Homer Simpson wins a fake award at a fake award ceremony set up by his employer to trick him into signing a legal settlement in lieu of suing because his workplace has affected his sperm count. The fake award is called the "Montgomery Burns Award for Outstanding Achievement in the Field of Excellence", and is presented by ex-boxer Smokin' Joe Frazier. Frazier uses the phrase in the "cliché" and/or "failing to act smart" way, poking fun at the meaninglessness of the award being presented. His whole speech is:

"Webster's Dictionary defines excellence as 'the quality or condition of being excellent.' And now the winner of the first annual Montgomery Burns Award for Outstanding Achievement in the Field of Excellence."

The Simpsons used a similar joke in the 1993 episode "Treehouse of Horror IV". Ambulance-chasing lawyer Lionel Hutz uses it in an unsuccessful defense of his client, Homer Simpson, trying to prove the contract Homer has signed with the devil is invalid. Here, the joke is of the "failing to act smart" variety:

"But I ask you, what is a contract? Webster's defines it as 'an agreement under the law which is unbreakable.' 'Which is un-breakable'!....Excuse me. I must use the restroom."

Realizing how unsuccessful this legal argument is going to be, Hutz sneaks out the bathroom window and abandons the court case.

The Simpsons reused the joke yet again in reference to marriage in the 1994 episode "Secrets of a Successful Marriage". In this episode, Homer Simpson falls backward into teaching a community college class on successful marriages, and the cliché is again used as a "failing to act smart" joke, with Simpson clearly unqualified to teach a college-level course:

"Now, what is a wedding? Well, Webster's Dictionary describes a wedding as 'the process of removing weeds from one's garden.'"

Homer has obviously misread the dictionary and given the definition for "weeding" instead of "wedding".

The U.S. version of The Office TV show used this exact same joke in the 2007 episode "Phyllis's Wedding", in the "failing to be smart" sense. Only this time, the character Michael Scott has mistaken the definition of "welding" for "wedding":

"Webster's Dictionary defines 'wedding' as 'the fusing of two metals with a hot torch.' Well you know something? I think you guys are two medals. Gold medals."

The wedding theme is quite popular. In the 2000 Friends episode "The One With the Nap Partners", the character Rachel uses it when she is unprepared to give a wedding toast for her friend, in the "impersonal" and/or "cliché" variety:

"Webster's Dictionary defines marriage as... Okay, forget that! That sucks! Okay? Never mind! Forget it!"

This same use, as a potential wedding toast, is found in the 2009 episode of Community "Urban Matrimony and the Sandwich Arts", where the character Annie essentially accuses the speaker, Jeff, of being cliché:

JEFF: "Webster's Dictionary defines..."

ANNIE: "Gah! Stop!"

JEFF: "Webster's Dictionary defines"?

ANNIE: "That's the Jim Belushi of speech openings. It accomplishes nothing, but everyone keeps using it and nobody understands why."

The Simpsons, too, has re-used the joke a couple of times, though outside the wedding cliché. In the 2003 episode "Special Edna", it's used by elementary school principal Seymour Skinner when presenting an award to one of the school's teachers, Mrs. Krabappel, who Skinner is dating. Here, it is used in the impersonal/long-winded and cliché sense, a play on Skinner's aversion to rule-breaking and risk-taking:

"As principal, I'd just like to say a few words about what this wonderful woman means to this school and to me. Webster's Dictionary defines Mrs. Krabappel as--"

The principal is interrupted and doesn't finish the sentence.

In the 2016 episode of The Simpsons "Simprovised", Homer Simpson is set to deliver a comedic speech at work that he has delivered annually, only to find out right before going on stage that the jokes won't work this year because they poke fun at a bunch of tragedies that have happened in the past twelve months (like poking fun at his co-worker Lenny's grandmother, who has since died). With nothing else prepared, he starts the speech off with the cliché, used in its clichéd sense, and also sort of the "failing to act smart" sense:

HOMER: "Webster's Dictionary defines... a speech as a... series of words that... (murmuring) elo... quently..."

LENNY: "I've never seen anyone bomb like that."

CARL: "Yeah. I really feel badly for the guy."

The TV show Children's Hospital has also re-used the joke in the "failing to act smart" sense, when a character mistakes the definition of "louver" for "love":

"Webster's Dictionary defines 'love' as a roof lantern or turret, often with slatted apertures for escape or smoke or admission of light in a medieval building."

Other TV comedy shows that have used it in a cliché, overwrought, or "failing to act smart" sense include Parks and Recreation, Family Guy, and Last Man Standing.

So where does the joke come from?

It is a bit cliché, but it's something that academics have historically done and continue to do. Not so much in speeches, but in written academic papers and books. Webster's and Merriam-Webster's are used sometimes, but it's probably more common to reference the Oxford English Dictionary. Depending on the subject at hand, it may be useful for the academic paper's/book's audience to know some definition of a topic that is going to be discussed at length as a starting-off point. This might be because the writer assumes the audience is unfamiliar with the word, or because the definition of the word might be integral to the argument that paper is going to be making. The Oxford English Dictionary is especially useful and is often cited because it attempts to catalog every word in the English language, every sense that each of those words have ever been used in, and the first instance in print of each of those senses, along with the full relevant quote from those cited sources, along with subsequent important citations as the sense of the word continued to be used further into history. This information might help bolster an academic argument being made. Papers that discuss the history of an idea, and when it first came to be used, will often find the OED indispensable as a starting-off point.

A quick search on Google Books and Google Scholar turned up many, many academic results into the present day for "Webster's Dictionary defines", "Merriam-Webster's Dictionary defines", "Oxford English Dictionary defines", and other variants, used precisely in the manner that the TV comedy shows are poking fun at. From the Google search results, some recent publications that use OED and/or Webster's definitions in this way include:

• Emissions Trading Schemes under International Economic Law by James Munro (Oxford University Press, 2018)

• Maintenance in Medieval England by Jonathan Rose (Cambridge University Press, 2017)

• Hamilton's Paradox: The Promise and Peril of Fiscal Federalism by Jonathan A. Rodden (Cambridge University Press, 2005)

• Wetlands Law and Policy: Understanding Section 404 by Kim Diana Connolly, et. al (American Bar Association, 2005)

• "Why Style Matters: A Comparison of Two Administrative Reform Initiatives in the Israelic Public Sector, 1989–1998" by Alan Peled (Journal of Public Administration Research and Theory, 2002)

• "Evidence-Based Practice and the Role of Nursing Leadership" by Cheryl B. Stetler, et. al (The Journal of Nursing Administration, 1998)

• "Perceptions and Misperceptions of Skin Color" by Stephen H. Caldwell and Rebecca Popenoe (Annals of Internal Medicine journal, 1995)

• "A Neural Network Model of Disturbances in the Processing of Context in Schizophrenia" by Jonathan D. Cohen and David Servan-Schreiber (Psychiatric Annals journal, 1992)

(cont'd...)