Not sure how up to speed you are with electrics in general, but i'll start it off.
The motor draws power from the battery. The speed controller is there to regulate how much power it will receive. It does this by limiting the amount of power that can pass trhough it to the motor. Kind of like a tap on a water pipe.
The current (amps) is a representation of how much power is passing through at any one time. Kind of like how large a water hose can be. A little hose can't pass a lot of water, a big one can pass lots.
So, if you have a high powered motor that wants to pull through a lot of power, you need an ESC that can allow this without too much resistence, or you will blow the seals on your valve.
If the battery has a lot of internal resistance, it won't let a lot of power through. Think of a water tank with a small hose coming out of it. If it has very low resistence, it can let a lot of power through, think bigger hose.
Hope that helps a little towards it.
BTW, I may have bits wrong, but there will be a fair few replies to correct me no doubt