The C rating is how the manufacturer describes the discharge rate of the battery. Although most batteries are made to the same specs they all perform differently after they are manufactured. This is the same with computer processors, all the processors in a given family are made on the same line, they are then tested after to see what their max rating is then thrown into the appropriate bin. The higher rated ones sell for more even though they are all made the same.

To get the amps from the C rating, just multiply the capacity of the battery in mah by the C rating then divide by 1000.

so a 20C 2000mah battery would be 20 x 2000 / 1000 = 40amps.

You're also right that you won't fry your esc or motor from a higher capacity or higher C rating battery. Power is PULLED from the battery not pushed. If you don't have enough amps is when you run into problems.