"The Day the Earth Looked Stupid" is very likely a reference to the Fallout video game series, as the whole city of Springfield is turned into a post-apocalyptic world after the alien attack, and the fact that the ending song is "The Ink Spots - I Don't Want to Set the World on Fire" which was actually never used in the game until Fallout 3, although it was intended to be used in the original Fallout but wasn't for copyright issues. The name of the episode is an obvious reference to the film "The Day The Earth Stood Still".
Sir Mix-a-Lot makes a parody of his best known song, "Baby Got Back" called "Baby Likes Fat" which plays when Homer starts eating people.
After Homer eats some fat German men at an Oktoberfest, he remarks "Must eat more fat people. Thank God I live in America."
In the segment "You Gotta Know When to Golem" The female Golem is made of Play-Doh! plasticine (as you can see Play-Doh! cans on the grass after Marge made her).
When Kang and Kodos invaded Earth, Kodos remarks that "we had to invade! They were working on weapons of mass Disintegration!" which is a reference to how American soldiers invaded Iraq because George H. W. Bush claimed they were working on Weapons of Mass Destruction.
The chaos in Springfield after the radio broadcast of H. G. Wells' "War of the Worlds" could be based off the mass hysteria in some parts of America and how many people died during the confusion.
The fact that "The Day the Earth Looked Stupid" lapses over three years is a commentary on, as many others have suggested, the US occupation of Iraq. Operation Iraqi Freedom, which later turned into Operation Enduring Freedom began in 2003; this episode aired in 2006, three years after the beginning of the occupation. Springfield, after internal hysteria, is invaded by an external force, and, therefore, is a model of Iraq. Kang and Kodos are, obviously, the American soldiers.
Originally, "The Day the Earth Stood Stupid" was supposed to end with one of the aliens commenting on how the abandoned Springfield looks the same way as Iraq will be in the future. While the FOX censors and executives had no problem with the line, the writers still had it removed, as they thought the reference was too blatant.
Halloween name for Al Jean was "Al "July 27th, 2007" Jean". This is a reference to Simpsons Movie release date.