How can I safely cut the cable on a broken garage door?
Unless you know how to unload the tension on the torsion spring, this is not safe for you to do yourself. There are plenty of videos on the internet that show you how to do it, but be aware that the consequences ofusing the wrong tools or doing it wrong will likely result in serious injury, such as broken bones or amputated limbs, or death.

You should probably hire a pro to unload the torsion spring.