The analogy of water flowing through a pipe can be really helpful here. Current can be analogized to rate of water flow at a given point in a loop of the pipe system (the circuit) and a resistor can be analogized to a particularly narrow part of the pipe, like a constriction. Assuming water has been flowing continuously for some time, the constriction will cause the rate of flow anywhere in the pipe system to decrease, and, similarly a smaller constriction causes less "resistance" and water flow, current, will increase.
Edit: This is not contradictory to ZapperZ's analogy. I just wanted to break it down further.
I think part of the misunderstanding is seeing current as it's own entity and not as a property of the whole loop within a circuit.
According to Ohm's Law, I = V/R, given an independent voltage source, reducing the total resistance "facing " that source will raise current flowing through that source.