I think it boils down to:

1. some places have good public transit, others don't
2. some trips just don't work well without a car, making people get cars
3. once you have a car anyways, it's often slightly more convenient

In some places, public transport is just bad, while car infrastructure is good. In some cities, on the other hand, public transport is great while going anywhere by car is slower than walking.

A car lets you comfortably get basically anywhere, anytime, in any weather, with any (reasonable) amount of luggage, usually on a more direct path than public transit. Nobody will bother you during that time.

Public transit:

• may not work if you have luggage
• may not go exactly where you want to go, requiring a lot of walking at the end
• may require significant waiting, including waiting for connections
• may take a lot longer, especially if you need to take some inconvenient connection because there is no direct one
• may simply not be available
• may be really uncomfortable or even unsafe (full, aggressive/rude/stinky people, dirty)
• may be unreliable

I don't have a car, and usually that works fine, except when it doesn't and I realize I'm missing out on something because to be there at 9:00 am I'd have to get up at 5:00 am, walk to the station, take the first train at 6:00 am, hope I don't miss any connections and sit on two trains and one bus vs. a 1 hour drive by car. Or I need to get to some place simply doesn't have any public transit in reasonable walking distance. Or I would like to transport two crates of beer.

These trips that just don't work on public transit make people get a car. Now they own a car and have paid the biggest part of the expense up front. And once you have the car, it's very often faster and more convenient to just get the car and drive somewhere rather than deal with public transit, which probably will take longer, won't go at the exact time you want to go, etc. - and most importantly, it requires a lot more planning and figuring stuff out than with a car.

Driving a car you own also appears deceptively cheap because maintenance etc. is often seen as a fixed cost (even though driving more increases the cost), so people only consider the cost of the fuel. Even with European prices, driving 100 km at 6 liters/100 km is like 10-12 EUR. A public transit ticket is going to be just as expensive, if not more. Especially if it's two people going together. So even if for some connection public transit may make sense, it can quickly price itself out of the game once someone owns a car.

Having car sharing easily available can solve the problem, but that's still a lot more annoying than a car you own, since it requires planning, you can get unlucky and not have a car available, and dealing with booking/pickup/return is a hassle. And it quickly gets more expensive than buying a really cheap car.