How does one become an official for bicycle races?
All of the information that you need is on this page for USA Cycling.

There are three levels for road cycling, starting with C, then B and A. C level I believe is just a clinic that you attend, pass the post test (70% minimum IIRC) and you must be 16. To move up from C to B, there are 11 modules that must be taken and passed, and then to move to Level A, there are requirements for races attended, level of races officiated, and other age requirements. The officials manuals detail these, and are located on this page.

There are different requirements for Mountain and BMX (I believe they skip the B level), and the local contacts for Missouri (If your profile location is still correct) are located on this page.

As far as fees, etc.,  that is going to vary from race to race and with the level of the race. I know at the races that our club puts on and a majority of the races in the region, the officials are compensated.