In Britain many play a game called 'fantasy football league'. You can do this online with thousands of people or you can run small competitions against your friends and work colleagues. To do this you are given a list of footballers with different prices and an imaginary budget. You then have to choose a team and you get points each week depending on how these players do in real life. What people in the UK call 'football' is called 'soccer' in the US and what people in the US call 'football' is called 'American football ' in the UK, but I imagine the game would work with both sports. Though organising a fantasy league is no 'sure fire way' to avoid 'getting fired'!

He’s saying he was the leader of the online fantasy football leagues so he couldn’t get fired as a joke. “I run ____” is a slang way of saying “I am in charge of _____” If someone owns a coffee shop, they’ll say “I run the coffee shop on the corner.”

I think that he's referring to Fantasy Sports, which is a popular thing in America.