Redress Control Number is used to help travelers who have been denied or delayed airline boarding; have been denied or delayed entry into or exit from the U.S. at a port of entry or border crossing; or have been repeatedly referred to additional (secondary) screening.
Airlines will modify their reservation systems to accept the 7 digits redress number to allow passengers with redress numbers to enter it during the reservation. This is a part of the Secure Flight Program.
Regarding how this works, the Redress Control Number (redress number) allows Secure Flight to match travelers with the results of their redress case to prevent them from going through the above mentioned scenarios. From Secure Flight Program website:
The airline submits this information to Secure Flight, which uses it to perform watch list matching. This serves to prevent individuals on the No Fly List from boarding an aircraft and to identify individuals on the Selectee List for enhanced screening. After matching passenger information against government watch lists, Secure Flight transmits the matching results back to airlines so they can issue passenger boarding passes.
If you have experience any of the above, You can apply for the Redress Control Number.