Can you tell us where you found this, longxianchen? I'm not familiar with this piece of slang, though perhaps someone else is. But it would help if we knew what variety of slang we're looking at, so the name of the book, the author, and a little bit about the character who's saying this would be useful.
A beat is a stretch of a river and, by extension, the right to fish in it. On a good salmon river, fishing rights can be very expensive.
Simply put, all salmon fishing rivers in Scotland are divided into individual stretches of river known as beats. Each beat can be 2 or 3 miles long and is determined by who owns the land on either side of the river.
I hadn't had a clue as to this usage of "beat," either, so it's been educational.
Is this related perhaps to a similar metaphorical-ownership usage of "beat" when talking about a police officer patrolling his area of assigned responsibility, "his beat", except in Scotland it's been (or has always been ) extended to mean actual legal fishing rights? Does anyone know if that usage is exclusive to Scotland? How about Ireland, since you say the author is Irish.