It may seem logical that the APU would start without the batteries. After all, the APU is able to charge some aircraft batteries.
With a closer look to the operation of the APU, it becomes clear that this isn’t true. When you start a car (or plane for that matter), the battery is powering a starter motor to get the components of the engine turning and the cycle going. Since the APU is just a baby engine, the battery is also required to start it.
The wise guys among you may be thinking, “how come lawnmowers and generators don’t have a battery?”. These engines also require motion to start. Rather than a battery, most of these engines have a pull string to get the engine turning.
Unless you want to pull a very long pull string, the battery is the simplest way to start up an APU.