It may be necessary to amputate a casualty’s limb in order to rescue them when:

A deteriorating patient is physically trapped by a limb where they will almost certainly die during the time to secure extrication

or

A completely mutilated non survivable limb retaining minimal attachment which is delaying extrication and evacuation from the scene

This paper provides an overview:

Porter KM. Prehospital amputation. Emerg Med J. 2010 Nov 13;27(12):940–2

and is summarised in this blog post.

These videos demonstrate the technique: