The Grey Poupon joke is a reference to a series of commercials from the 1980s and early 1990s, which depicted affluent, older men in limousines asking each other if they had any Grey Poupon mustard, a product marketed as a luxury condiment. The humor in the “Wayne’s World” movie arises from parodying this ad, evoking the commercial’s association with high-class tastes in an irreverent and unexpected setting​.
Here is one of the original commercial:

The joke “Why did the bird go to the hospital? It needed a tweetment” is a playful pun that combines bird behavior with medical terminology. The humor arises from the wordplay involving “tweet,” a sound commonly associated with birds, and “treatment,” a term for medical care. By merging these words, the joke creates the term “tweetment,” humorously implying that the bird requires a special kind of medical attention unique to its nature. This pun is amusing because it anthropomorphizes the bird, attributing human-like healthcare needs to it. Additionally, the joke plays on the similarity in sound between “tweet” and “treat,” which adds to its linguistic cleverness.

The joke “What do you call birds that stick together? Vel-crows!” is another example of a pun, where the humor arises from the play on words. This joke blends “Velcro,” a well-known brand of hook-and-loop fastener known for its sticking properties, with “crows,” a type of bird. The humor comes from the clever manipulation of the word “Velcro” to sound like “vel-crows,” suggesting an amusing and imaginary scenario where crows have the ability to stick together like Velcro. This creates a whimsical image of crows behaving in a manner that defies their natural behavior, sticking to each other as Velcro does.

It’s funny because jokes about eating crayons are occasionally made by Marines as well. Much of it stems from the tasks that Marines are frequently assigned, which call for the ardent destruction of property and the people who protect it in a manner similar to a toddler destroying a room—and the fact that Marines are exceptionally skilled at this.
In addition, a great deal of the duties assigned to the Marines are rather simple tasks. When people from other services refer to Marines as crayon-eaters, they usually mean that they behave like that destructive kid again—that is, they do stupid things with great passion.

The joke “We all know Marines eat crayons, but what’s their favorite flavor of crayon? Crayonberry” plays on a stereotype and wordplay for comedic effect. Firstly, there’s a humorous stereotype or running joke suggesting that Marines, known for their toughness and no-nonsense attitude, are so straightforward that they would eat crayons, an item typically associated with children and non-edibility. This stereotype is used here for light-hearted humor, not to be taken seriously.
The punchline “Crayonberry” is a play on words, combining “crayon” and “cranberry.” Cranberries are a common fruit flavor, and by blending it with the word “crayon,” it humorously suggests that if Marines were to have a favorite crayon flavor, it would be something akin to a real fruit flavor, but still within the realm of crayons.

The joke “What did one potato chip say to the other? ‘Shall we go for a dip?’” employs a pun, a type of wordplay where a word or phrase has a double meaning, creating humor. The term “dip” in this context has two meanings. Firstly, it refers to a sauce or creamy mixture into which snacks like potato chips are often dipped before eating, enhancing their flavor. Secondly, “dip” can mean to go for a brief swim or plunge into water. The humor lies in the dual interpretation of “dip.” In the literal sense, potato chips can’t swim, but they are commonly paired with dips like salsa or guacamole.