The Little Black Box is a small bar on the grounds of the Springfield International Airport, and is exclusively for pilots and those who have played one or the other on TV.
Being kicked out of Moe's and not finding any other good bars, Homer goes there, pretending to be a pilot. When the airport needs a pilot, they picked the one who wasn't drunk, and that was Homer (as he hadn't even gotten a chance to get a drink yet before they made the request).
Behind the Laughter
The name of the bar is a pun on a "black box", a device found in aircraft.
The Little Black box is now renamed to Star and is owned and managed by Aston Merrygold and Waitress Cloe.