Because it is actually : Bill Board
People would receive various notifications that they would refer to as 'bills'.
Today we have separated that term into multiple meanings. Such as a 'flyer'.
The board part is self explanatory I guess.
Old: I got a bill in the mail today about a sale
New: I got a flyer in the mail today about a sale
Electric Bill, Phone Bill, etc. All Bills is drawing attention to something.