You get the license from the BATFE. You get the application, ATF Form No. 8from their website, fill it out completely, return with the $30 fee, and send a copy to your local sheriff.
Unless the local cops object to your having such a license, the BATFE checks you out, if all is well, you get your license in 30 days or so.
The license allows you to purchase C&R eligible firearms without further 4473 forms or NICS checks, although many dealers still do them. Theoretically, you can furnish a signed copy of your license to a dealer, pay for the C&R gun, and walk out with it. Reality is usually different. You can order C&R guns over the internet or by mail, and have them shipped to your door. Non-C&R guns you still buy from a dealer in the usual manner.
You keep a record of all your C&R purchases, during the time your license is current, and record all transfers or disposals of C&R items from your collection. BATFE can perform audits of licensees, but only a few cases have been discussed on this board, so it's rare. They have other things to do. At renewal time, you state how many C&R guns have been purchased or sold under your expiring license term, a large number sold may get some attention. It is not a dealer's license, it allows you to engage in collecting activity, not dealing for profit.
That's it in a nutshell.