Initially when you attach the capacitor to the battery, said battery will act to create an electric field within the wire. On the side of the negative terminal this field will point perpendicular to the cross section of the wire toward the terminal of the battery (electric field points toward negative charge). On the side of the positive terminal the field will point perpendicular to the cross section of the wire away from the terminal of the battery. This is because the battery arranges the electric charge in such a way that the negative terminal is electron rich and the positive is electron deficient.
Now, this electric field will cause charges to move. On the side of the negative terminal, the electric field will push the electrons away from the battery and toward one side of the capacitor. On the side of the positive terminal, the field will pull electrons from the capacitor and toward the battery. When this happens we see that the side of the capacitor attached to the negative terminal of the battery is accumulating electrons (becoming negative), and the side of the capacitor attached to the positive terminal of the battery is losing electrons (becoming positive). But what is this motion of charges doing to the field?
With electrons accumulating on the negative side of the capacitor, the field within the wire is decreasing (because now there is less of a potential difference between the negative terminal of the battery and the negative side of the capacitor due to the accumulation of the charges). Electrons will continue to be pushed toward the negative side of the capacitor, however, until this field decreases to zero. This will happen when enough charges accumulate on the negative side of the capacitor to bring the negative capacitor plate to the same potential as the negative terminal of the battery. By the same argument, electrons will continue to be pulled away from the positive side of the capacitor until the field is brought to zero when the positive capacitor plate achieves the same potential as the positive terminal of the battery.
From reading the comments above it seemed to me that this is the type of explanation you are, perhaps, looking for. I hope it is helpful.