How Does DC Current Charge a Capacitor?
Imagine two metal plates very close together, but not touching.  They are so close that the electrons on one plate can 'feel' electrons on the other.

When you apply a voltage, you are pushing electrons onto one plate, which repel the electrons on the other plate.  You can imagine that for every electron you push onto one plate, an electron leaves the other.

This happens until the voltage is no longer sufficient to push anymore electrons onto the plate.  This happens because you have pushed so many electrons onto that plate that the force of them repelling each other is equal to the voltage you have applied.

In a DC circuit, the charged capacitor is similar to an open switch as no more current will flow.