At a bottleneck, the water that gets through is moving faster, but the total volume of water going through is lower.
It's a classic EE analogy, so people tend to shorthand it. One good and thorough explanation of the full bit is here: http://science.howstuffworks.com/environmental/energy/questi...
What happens if you increase the pressure in the tank? You probably can guess that this makes more water come out of the hose. The same is true of an electrical system: Increasing the voltage will make more current flow.
Let's say you increase the diameter of the hose and all of the fittings to the tank. You probably guessed that this also makes more water come out of the hose. This is like decreasing the resistance in an electrical system, which increases the current flow.