The previous answers actually describe the formation quite well, but I'd like to go into the actual dynamics a bit more. It's not just about the air layers and wind shear - what's crucial is that a rotating column forms from a thunderstorm, a kind of mesocyclone. That happens when horizontal wind shear (meaning the difference in wind speed and direction at different altitudes) gets converted into vertical rotation by updrafts. A vortex like that can become a supercell in just a few minutes and spawn a tornado. The difference from a normal thunderstorm is exactly that: a thunderstorm is spread out, moves along, and causes heavy rain and lightning. A tornado is this concentrated, rapidly rotating tube - super localized, but also super destructive.
When it comes to the danger, it's less about the formation itself and really the combination of extreme wind speeds (some tornadoes reach over 300 km/h), the unpredictable path, and the fact that they form and disappear quickly. You don't have time for major evacuations. That's why warning systems in the US are so important - radar systems can detect the rotation before a tornado even touches down.
For protection: in the US there are tornado shelters or basements everywhere. If you don't have a safe place to go, you should get to a bathroom or an interior room with no windows - stay away from outside walls. The most important thing honestly is to take the warning systems seriously and not hesitate. In Germany you don't really need to worry about that much - actual tornadoes are extremely rare here.