Although we don't require travel documents to travel within the country, many disaster areas will be restricted to residents for a certain time period. A driver's license is probably the easiest, most universally accepted proof of being a resident (in the US) of some disaster area, like after a flood, tornado, wildfire or other calamity.

I remember watching the TV news of a wildfire down here not long ago. People were already at work when the wildfire broke out and lot's of people sped home to gather belongings, get pets, etc. Only residents were allowed back into the threatened neighborhoods because the flames were very close at that point and it was dangerous to be there. When minutes counted, it was easiest to flash the driver's license with your address than to try and scrounge up other documents that have your address on it. For privacy or other reasons, some of us have PO Boxes or other alternate addresses on all of our bills, our driver's license, on the vehicle registration and insurance card which is different from where we actually live. I'm not sure what happened to people at that wildfire who couldn't immediately prove their addresses, but I'm pretty certain they couldn't just drive right in like the others.

There could be curfews within disaster zones, particularly if there are already any problems with security, like reports of looting. If you're stopped by the police, you're probably far less likely to be hassled and grilled if you can prove that you live just a couple blocks down the street than if you live in the next county.

Proving your identity is important, but in some situations, so is proving your residency.