In the case of a 30A ESC, thats as much as it can ever handle without letting the smoke out. If your motor is liable to draw anything like that off your chosen voltage, you need a bigger ESC or a motor that draws less current (and therefore delivers less power).
To produce a high level of performance, forget SLA. They are good for displacement hulls that need ballasting for stability and a moderate amount of current for a long run, not for tearabouts.
Ampere-hours are the electric equivalent of gallons, the bigger the number, the more juice in the tank.
Amperes equates to the rate at which the fuel is delivered. The bigger the pipe, the more fuel you can get through it in a given length of time.
Volts is like the pressure of the fuel pump - the harder it pumps the higher the pressure (voltage) and the more fuel can be pushed through the pipe.
Some batteries can deliver huge currents for a short time without damage. SLAs are not amongst them. The plates tend to heat unevenly, go curly, touch, short the battery out, and kill it.
If the brushless unit takes more current than the supply can deliver, the voltage will drop. If this is supplying the control circuitry, this will stop, causing the motor to stop. If the voltage then comes back up, the cycle might restart.