Why is it dangerous if air gets into a car braking system?

Modern vehicle braking systems use hydraulic brakes. This means that the system is filled with hydraulic fluid. The fundamental principle of hydraulic science is that this fluid is not compressable. Air is compressable, so if air gets into the brake system, the air will compress when you step on the brake pedal, instead of the car stopping. If there is only a small amount of air in the system, the pedal will feel "soft" or "mushy", but the car will still stop. However, if a significant amount of air gets in the system the pedal will simply go straight to the floor and the car will not stop. While air in the system is very dangerous, it is easy (and free if you can do it yourself) to fix. Just bleed the air out of the system. A manual can outline how you do this correctly. It takes two people, a few basic tools, and about an hour (depending on your skill) to do.
Hydraulic brake systems use the fluid to transfer the force from the brake pedal to the braking system. Fluids can not be compressed. Air bubbles can be compressed and will make the braking system less effective.
In hydraulics, we rely on the ability of the hydraulic fluid to transfer energy from the input end of the system to an output end. If the energy associated with compression of the fluid at the input end is "used up" compressing a bubble in the line, that is just that much energy that won't get to the output end. Looked at another way, compression of the fluid at the input end results in compression of the bubble rather than the delivery of energy to the output end. In a brake system, this is a disaster. It leads to poorly performing brakes, or even a failure. On a moving vehicle, the failure of the brakes frequently spells disaster.