There are people who do not believe that identity theft is a problem. Sadly a person does not have to believe that a type of crime is possible to become a victim of it. Identity theft is a first world modern kind of problem that can not be solved just by locking doors.
Once identity thieves have your personal information, they can drain your bank account, run up charges on your credit cards, open new utility accounts, or get medical treatment on your health insurance. An identity thief can file a tax refund in your name and get your refund. In some extreme cases, a thief might even give your name to the police during an arrest.
The Federal Trade Commission has tips on how to prevent and how to detect identity theft and fraud.
Personally I just love my paper shredder and the online banking services I have where I can set up alerts and even limits as to how much money someone can take out of my accounts. I have not carried anything on my person that has a social security number on it for decades now and I keep my phone and tablet locked and passwords on apps that access sensitive information so that if someone does unlock one of my mobile devices they still can not access any private information.