How to root Galaxy S3
How to root Samsung S3
Common FAQ about process:
Rooting process
Rooting on OSX Odin flash guide
Basic instructions
All downloads contain .img files bootable through Fastboot.
Make sure your device is in Fastboot mode (usually you can get there by holding VolUp+VolDown+Power when the device is turned off), then run the included script On OSx, you may have to chmod +x mac.sh before the script does anything.
A special boot image will load, which will install (only) SuperSU, thenreboot into Android