The story
From furry (“relating to the furry subculture”) + number, referring to the website e621.net. The website's name, e621, refers to monosodium glutamate, a chemical used to enhance taste in food, referring to its pornographic content being enjoyable ("tasty") by fans. Compare sex number (“69”) and weed number (“420”).
Etymology from Wiktionary, CC BY-SA 4.0.