What it means
Means something was done honestly and properly, with no cheating, no sneaky edge, no funny business. You say it when someone really won clean, or when it's time for the loser to stop moaning and hold the L with a bit of dignity.
Usage examples
"Alright mate, you beat me fair and square, no dodgy ref calls. Next round’s on me, but I’m having a rematch after this pint."
"She won the match fair and square, no lucky bounces or dodgy calls."
"We split the bill fair and square, everyone paid exactly their share."
"You got me fair and square, mate. No excuses from me, that finish was tidy."
"He nicked my parking spot yesterday, but in cards I cleaned him out fair and square, so we're calling it cosmic justice."
Where it comes from
The phrase is old English and has been around since at least the 16th century. Fair keeps the sense of honest and right, while square was long used for something proper, straight, and even. Put together, it landed as a neat little stamp for clean dealing and a result nobody can really argue with.