This guide will let you do exactly that, with step-by-step easy to understand instructions to root your AT&T Galaxy S3. Rooting lets you access system files, perform modifications, overclock, etc, and use all the apps that require root access to work, and doing it is quite simple as long as you follow all the steps carefully.
Copy the CWM_SuperUser_v3.0.7.zipfile to the internal SD card on the Galaxy S3. Do not extract the zip file, copy it as it is to the internal SD.
Now, boot into ClockworkModrecovery. To do that, turn off your phone. Then, hold downVolumeUp, Home and then the Power buttons together until the screen turns on, then let them go. The phone will boot into ClockworkMod recovery in a few seconds. In recovery, use the volume buttons to scroll up/down and the home button to select an option.
Select install zip from sdcard, then select choose zip from sdcard. Then, scroll to the CWM_SuperUser_v3.0.7.zip file and select it. Confirm installation on the next screen by selecting Yes.
After installation is complete, press the back key to go back to the main recovery menu, then select reboot system now to reboot the phone.
Now, after the phone boots up, we need to install BusyBox on the phone. To do so, follow the steps below:
Install BusyBoxon the phone from the Play Store here
Run the BusyBoxFreeapp on the phone.
Allow it root access when it prompts you, by tapping the Allow button.
Tap the red “X” to close the dialog window that pops up.
Allow BusyBox to scan your phone.
Tap Install to install BusyBox and make the necessary file changes. After installation is done, close the BusyBox app.