What is the function of a battery? One may think that it is to store energy or create electricity, but the primary function of a battery is to maintain a constant potential difference between its terminals: the positive terminal at a higher potential, and the negative terminal at a lower potential. The difference between them is called the battery's emf \({\cal E}\), and is what you'll find printed on batteries you buy in the store: AA batteries have an emf of 1.5V, 9V batteries have an emf of 9V, car batteries have an emf of 12V, etc.
When the terminals of the battery are connected by a piece of metal, such as a wire, a competition ensues. The metal, being a conductor, wants to maintain a constant potential between its ends. The battery, on the other hand, wants the top of the wire to be at a higher potential than the bottom of the wire. The battery wins, in the short term, by continually pumping charge from one terminal to the other to maintain its desired potential difference. Once the positive charge reaches the positive terminal, it then flows "downhill" through the wire to the negative terminal. This continual flow of charge is called an electric current. The battery requires energy to pump charges uphill, which it gets from chemical processes that go on in the battery. When the chemicals have completely combined, the battery runs out of energy, and then the potential difference between the terminals returns to zero. The battery wins the battle, but the wire wins the war.
Instead, we can ask "why are the charges moving?" The battery sets up a potential difference, and positive charges are moving from higher potential to lower potential: that is, downhill. And in Section 7.6 we said that the electric field points downhill as well, so the electric field must be pointing in the direction the charges are moving. Or from another point of view, if we ask "what is pushing the charges through the wire?", clearly it must be an electric force. Positive charges feel a force in the same direction as the electric field. Thus the electric field at the star points downward.