There are several ways. Firstly when the car is hit in the side it suddenly starts accelerating sideways. If the people inside the car is not wearing seatbelts they will not be accelerating with the car. And so the car door will come and smash them in the side so hard that the glass shatters and the steel doors buckle and tear. The door might therefore let people through it, either through the window or through a torn up opening. It might for example just catch a leg and tear it off as the rest of the body is squezed through the window hitting the other car.

But if the car door is able to keep the passengers inside the car they are now bounced back inside the car. They might also hit things like the center console, dashboard, car seats, etc. and get bounced all over the place. If someone hits a weak point in the car such as the windows, windscreen, sunroof or a damaged door they might do so with enough force to go thorugh. Not always the entire body though, commonly only an arm, a leg or a head is able to get outside and the rest of the body stay inside the car. There are a lot of forces involved here.

And lastly as you say the car may spin out of control, either sideways or rolling. This creates a centrifugal force pushing everything outwards. So again people might get smashed against windows and doors already weakened or broken from the initial crash.

The seatbelts are there to protect you. They keep you in your seat away from any of the windows and doors as well as away from any impacting vehicles. Instead of getting smashed in the head by the inside of a car door which used to be on the other side of the car a fraction of a second ago you get a comperatively slow acceleration with the car.