Put simply, the engines work most efficiently at a certain number of high revs per minute. If, in your example, you remained at FL350, as the mass of the aircraft burned off, to maintain your cleared Mach number, you would have to reduce power into a relatively inefficient regime. In order to get the engines back into their most efficient revs per minute, you request climb clearance into thinner air.
There are other factors to consider, of course. There's no sense climbing into a greatly increased headwind component which would cancel out the benefit. That's where the job still calls for a bit of the cerebrals in making the decision.
