To laugh coarsely or loudly, especially at the discomfort of others.

This is a UK phrase, from around the time of WWII. It is first recorded by Eric Partridge in A dictionary of forces' slang 1939–45, 1948. He describes it as 'Ward-room and also Army officers’ slang'.

The reason why drain was picked for this simile isn't clear. Most similes include items that especially display the property being described, e.g. as white as as snow. Drains don't immediately make one think of laughter, although the gurgling sound might have been thought of as being similar to chuckling.