A very common misconception about electrical systems is that current is pushed into the motor. It is not. The motor pulls what it needs.

Think of the battery as a fuel tank. You wouldn't worry about blowing up an engine just because you put a bigger fuel tank on it would you? The only difference is that it could run longer. With some exceptions (discussed below) the situation with electrical power systems in RC cars and planes is the same. More battery capacity and multiple batteries run in parallel don't change what happens to the motor because it will only draw the amps it needs to run the speed that the voltage it sees dictates that it should.

However, in the real world changes to any component can have unintended effects on other components. Your motor will run the speed that the actual voltage it sees says it must run. Unloaded, that is simple enough. If the motor sees 8 volts and runs 1000 rpm per volt (known as the Kv rating) it will go 8000 rpm and draw as much current as needed to do so. When a motor is under load, it will pull more amps to keep the speed up. But because more amps are being pulled from the battery, the voltage will drop due to resistance. This resistance comes from every item in the electrical system and can make a significant impact. So maybe the motor under load only sees 7 volts and turns 7000 rpm. You can see this easily enough when you put a bigger prop on and can hear that the motor has slowed down a little.

So here is where things can get a bit tricky. If you buy better batteries, use bigger wire or better connectors, use multiple batteries in parallel or increase the battery capacity, you will reduce the resistance in the system. So the voltage under load will go up, speed will increase, and the amp draw will increase. And going to bigger batteries also adds weight, which then increases load and therefore amp draw. RC modelers usually allow a 20% or so margin of safety in our electrical systems so that those factors won't matter. But if you are running your power system right on the edge of what it can handle and then make a battery change, that can make the difference between survival and not.