The first thing that happens is that you send in the registration and receive your certificate for your achievement. Members who earn ATM's are listed in Hall of Fame section of the Toastmaster magazine, and TI World Headquarters will send a letter to your employer if you wish notifying them of your accomplishment.
After earning an ATM Bronze, you have two courses of action open to you. One is to continue giving advanced manual speeches, and work towards the ATM Silver and ATM Gold awards. The other is to seek leadership opportunities and earn the CL and AL awards. The paths are NOT mutually exclusive; members are encouraged to continue working down both avenues as they continue their membership in Toastmasters.