Taxis are expensive and slow. There are relatively few of them, so you spend 20-40 minutes waiting for them to pick you up, and then pay them 5x-10x as much as it would cost you to drive yourself.
If self-driving cars were even half as common as regular cars, there would be a dozen parked on the street by your house. You wouldn't need to wait more than a few minutes, and cost could be lower than standard car ownership due to better utilization of the vehicles.
The whole point of mentioning taxis was that the part about self-driving cars where you don't have to control the vehicle and where the vehicle just goes and picks someone else up after you are done with it is already served by taxis. It's definitely likely that self-driving cars will displace taxis, but it's not clear that it will actually be all that cost-effective in most places. The argument to be made is that the driver is such a large fraction of the cost of a taxi that reducing that to a single fixed-cost addition of a computer + sensors is going to dramatically change the pricing structure in such a way that people will give up personal vehicles.
I think that's very likely not true in most places where population densities are low. There will probably be some rebalancing, but I don't think it will be as dramatic a shift as the parent comment made it out to be.
I'm in a town in a ~2,000 square mile county, where the county population is less than 40,000 and the town is a bit more than 10,000. There is a county transit authority (which is subsidized) that runs a couple of buses and a few taxis (I don't think they are subsidized, but labor here is cheap and it isn't a hard place to find your way around).
There's lots of places with lower population density than that, but the viability of something that costs 60 or 70 percent of what a taxi costs kicks in at some pretty low point.
If self-driving cars were even half as common as regular cars, there would be a dozen parked on the street by your house. You wouldn't need to wait more than a few minutes, and cost could be lower than standard car ownership due to better utilization of the vehicles.