Why do mama bears protect their cubs?

The same reason any animal protects there offspring. Maternal instinct.

The world is a dangerous place for baby animals. Unless enough babies survive into adulthood, the specie would die out.

Animals have two ways of dealing with this, either have lots of babies, and hope that some survive by themselves. Fish and reptiles usually work this way.

Or the mammal approach, where a considerably smaller number of babies are born and then protected and cared for until the have a decent chance of managing on their own.