> Why should one need to buy a liability insurance policy on a self-driving car?

The same reason property owners in general by liability insurance for liabilities that may occur to do property they own.

> What driver exactly am I insuring?

Most likely, you're insuring against liability resulting from your obligation as the owner (or leaseholder-in-possession, in the case of a lease) of the autonomous vehicle to maintain the vehicle in condition for safe operation and to remove it from operation if that is not possible.

The liability in car insurance is virtually unlimited. Cars are very good at killing people and damaging property. You don't have to insure against small payouts (large excess gives you small premiums), and this is an area where the insurance companies want to upsell you.

The risk you're insuring is negligent or wilful bad driving, and risks inherent in the technology.

All of the above is true for self-driving cars, only with much-reduced negligent risk. The premiums will go down significantly, but the consumer motivation and the public policy interest to have insurance will remain. It may be assumed by the manufacturers for 100% self-driven cars, but insurance of outsized payouts is not something unique to the current iteration of the car industry.