Accidental Aesop: The writers admitted there was no message in the episode beyond "Idiots like Homer Simpson should not own guns". However, many have found their own Aesops from the narrative.
The importance of gun safety. The problem isn't that guns are bad, but rather guns are need to be handled responsibly and irresponsibility can lead to messy consequences.
Guns are bad. Regardless of Homer's portrayal, the when the NRA has an Everyone Has Standards moment it's played for Hypocritical Humor with Cletus (a dumb trigger-happy redneck) and Louis (a violent gangster) accusing Homer of being stupid and dangerous respectively. That combined with Moe joking about shooting a young robber in the spine casts a harsh light on the kind of people that own guns which the creators didn't intend.
Americans Hate Soccer: The soccer match at the start of the episode is perceived as boring. This is a joke that could only be made in the USA where, in contrast with the rest of the world, soccer has never caught on. Word of God said they were skewering how soccer was more boring on television than live.
Bart and Milhouse playing "William Tell" with the apple in Milhouse's mouth (and then jinxing Bart).
Marge's remark about how it's hard to believe that the soccer stadium used to be an internment camp.
The Sarcastic Middle-Aged Clerk's gun shop also carrying sympathy cards.
The montage of Homer waiting for his gun, showing all the things he could be shooting right now: bunnies, baby ducks, a truck carrying merchandise for Target, his sisters-in-law (Patty and Selma) on a tandem bike, and his overly-cheerful neighbor (Flanders) on a riding mower.
In the DVD commentary for this episode, Yeardley Smith admitted that Nancy Cartwright was better at doing the voice vibration technique during the scene where Bart and Lisa were having a race on their vibrating motel beds.note Which is done by poking your fingers against your throat while talking. Two years after the episode originally aired, another voice actor would give another popular yellow cartoon character his iconic laugh by using the same technique.