I had exactly the same problem on my 2007. When I turned the key to the "on" position, I heard something spinning. The "something spinning" is the fuel pump. I also saw the dash needles doing their happy dance. Pressing the starter button gave me: nothing. Shorting the starter relay got the starter working vigorously.
First things first: 1) Make sure your transmission is in "neutral". 2) Make sure the red engine cut-off switch is in the "Engine on" position.
I replaced the starter relay with a new one purchased on Amazon: (MOTO1988 Starter Solenoid Relay for Yamaha YZF-R1 YFZ450 FJR1300 XV1900 5JW-81940-02-00): $19.76CAN.
This was an easy swap and the new one fit exactly. The engine started up fine (once I replaced the spark plugs, but that was a different problem).