Consider the bathtub-bucket image below.

The bathtub on the left is connected to a bucket through a valve.
When you open the value the water will flow from the bathtub to the bucket. The rate at which the water flows, starts out fast and exponentially decays as the bucket reaches the same height as the bathtub water.
This is the same as a battery and a capacitor. The Battery is the bathtub and the capacitor is the bucket. Electrons are the water. When you close the switch, electrons move from the battery and "fill up" the capacitor till it reaches the same voltage as the battery.
The water does not go "through" the bucket, but it DOES go through the pipe. In the circuit, the is no current through the capacitor, but there IS current in the wires between the battery and the capacitor until the capacitor is fully charged.