Do you remember the last time you worked hard on an essay or paper, only to lose it all when Word or your computer suddenly crashed? Hours of work are gone because you didn't hit the save button often enough. That sort of frustration could drive even the bravest of souls to tears.
You might be able to afford this kind of data loss once in a while, but now imagine it's an entire database with Terrabytes of information—not just one document. Especially if that information is in a production environment, the damage could be catastrophic, costing millions of dollars. Not even mature organizations can afford these mistakes, and database backups prevent precisely this kind of situation.