How many "safety" schools should I apply to? And what does the term mean?

Safety schools are those you apply to knowing that you're practically guaranteed admission because either you greatly surpass their admission criteria or their admission rate is pretty high. Applying to one or two safety schools that you would actually like attending is a good idea in case complications arise with your dream school.

Depending on the price you would pay for tuition, your safety schools might be a better choice than the more prestigious schools.

I always wonder why people don't just apply to their safety schools and choose from the most prestigious of those, once they get their acceptance letters.

A safety school is a school where you are pretty much guaranteed admission. Choose carefully, and you should only have to apply to one safety school.