It means that when the motors start up from standstill, they will each draw 60A. Also if the motor is prevented from rotating, it will try to draw 60A.
I suggest you choose a motor controller with built-in current limiting, to constrain the startup current to a more reasonable value. It's possible to use the Arduino itself to do the current limiting if you choose a motor controller with current sensing, but then you need to get the software right or you risk damaging something.