We stated before that E = 0 inside a conductor, but that applies at electrostatic equilibrium. If you connect a battery to a wire, on the other hand, the battery will maintain a potential difference between the two ends of the wire. In that situation there will be a field in the wire, and charges will flow (and keep flowing) in response to this field.
Our picture of the charge carriers in a conductor is similar to an ideal gas. Charges (often electrons) move about randomly, scattering off atoms and molecules. When there is no field in the conductor there is no net flow. Applying a potential difference, which sets up a field, is much like creating a pressure difference in a gas. In both cases there will be a net flow.
For the charge carriers in a metal, the applied field superimposes a net drift velocity vd onto the random motion already there.
If the colored balls represent conduction electrons in a metal, in which direction is the applied field when the field is turned on?
If you turn on the field and watch the purple ball (which represents a typical electron) for a while, you should notice a net drift to the right. The field must therefore point to the left, because that would generate a force to the right on negative charges.