How could a knife wound in a fight cause amputation of an arm by surgeons?

Loss of Circulation. A knife wound can sever important blood vessels and muscle tissues. Infection of the site may further damage the area, leading to gangrene or life threatening infection. Sometimes amputation is the only viable option and may be life saving.
Blood flow. If blood flow to the arm is seriously affected, function is severely compromised, or infection sets in an amputation may be necessary in spite if best efforts. The procedure can be life saving.