According to his website, Day is a stand-up comedian. This explains the high ratio of amusement in the show.
"A Daily with no story," from Day Daily #337 Part 2 (starts at around 14:30). And I quote: "We are going to watch this game and, with not a single word spoken, we are going to receive... a story. Ladies and Gentlemen, I present to you: grenelve vs. Goose."
To explain, the Funday Monday challenge is for a Terran to immediately lift-off and move somewhere. It's TvT, and Goose kills one of his six workers, loads the other five up, and flies his CC off the map. Then he says, via chat, "[Proxy Error #404: Disconnecting]". His opponent sends a worker out to scout as normal, sees no base there, and then... disconnects, scoring Goose the win.
In one of the Day9 Dailies Funday Monday episodes, Day9 is commenting when one of the players, after hearing that this was for a Funday Monday, says that he hates Day9 for ruining Christmas for his daughter. Day9 looks appalled at this and is categorically denying that he did anything to his daughter and is utterly incensed at the accusation. Then the guy elaborates, saying he was watching one of his casts with his daughter and Day9 cracked a joke about Santa Claus not being real, which utterly destroys this little girl. Cut to Day9 looking sheepishly into the camera.
From Day9 Daily #381, though you'll only get it if you're knowledgeable on college-level math:
The inches of my penis form a countably infinite set, because when you think your out of more inches you can simply find more by inspection.
If you draw a curve of her pleasure in bed with me, she gets asymptotic close to climax. And right when she is like "I'm epsilon there, I epsilon there!" thatís when I climax, and you see it breaks off like a step function. Because at that point sex is no longer continuous.
Funday Mondays #293 and #381: The Worst 2v2 Partner Of All Time. So popular, it meritedreprisinglater. The only place you'll see a match where one player uses a 6-pool tactic to kill his ally, while one on the other team takes it to mean "build six spawning pools" and is left with precisely 0 Worker Units and 0 minerals.
Funday Monday #551: Day's rather (accidentally) suggestive Nydus Worm impression, and the request to the community not to 'shop inappropriate GIF images from it. Day himself begins laughing hysterically when he realises he was "practicing for Valentine's Day".
Funday Monday #604: Mind Games with ItMeJP. Day and JP are basically just messing around, doing their placement matches in the most bass-ackwards way possible, when Sean's sound cuts out. The rest... you must see to believe.