From the TSA website
We understand passengers occasionally arrive at the airport without an ID, due to lost items or inadvertently leaving them at home. Not having an ID, does not necessarily mean a passenger won’t be allowed to fly. If passengers are willing to provide additional information, we have other means of substantiating someone’s identity, like using publicly available databases.
Passengers who are cleared through this process may be subject to additional screening. Passengers whose identity cannot be verified by TSA may not be allowed to enter the screening checkpoint or onto an airplane.