My son just bought a Mazda. The floor is much lower than the part, (threshold?) I have to step over in order to get in. Why?

The bit you described is called the "sill". It forms a major structural part of the car, and deep sills are usually stiffer than shallow ones. 

Is the car a Miata (also known as the MX-5 or Eunos depending upon which country it was first sold in)?

If so, then that car was designed from the start as a convertible, so without a solid roof to act as a structural component the sills have to do that job. If they were shallow then the car body would flex across the middle, the doors wouldn't always open and eventually there would be fatigue cracks. 

Many other convertibles are developed as an afterthought by chopping the roof off a regular car. They then have a lot of added weight because the existing sills have to be given a lot of additional reinforcement and even after all that those cars often creak and shudder over bumps and when cornering because the body shell still twists a lot more than is desirable. That is an unwanted behaviour called "scuttle shake".

The Porsche Boxster is one of the few other current examples of a car designed from the start to be a convertible but manages to avoid having very deep sills due to the superior design which a far more expensive car allows for at the design stage.

it's to keep you from falling out while you are driving

Imagine what would happen if the floor was flat. 

Can't imagine? 

Put a piece of plywood in there and find out empirically. 

Because that's the way the body was made jojo.

Most cars are like that. Most trucks are not. Some of it may have to do with a uni-body vehicle versus something that sits on a steel frame. They do things with the sheet metal to add strength and of course styling.

Hudson introduced the "Step Down" floor on their 1948 models (to make the vehicle lower without reducing interior space) and they were adopted by the rest of the industry within a few years.  The last cars with flat floors I can remember were Studebakers thru 1966 and Checkers to their end in 1982.  How old was your son's previous car?