According to the Field Safety and Training school of the United States Antarctic Program the primary causes of death in a crevasse are:

The fall. A crevasse can be tens of feet, to over a hundred feet deep. Falling can cause broken bones or internal injuries that result in death either immediate or delayed. The cold. The temperature inside a crevasse is often lower than that at the surface. Temperatures as low as forty degrees below zero (C/F) are common. The victim will not have freedom of movement or will be injured and so cannot generate body heat as effectively. In addition, the victim may not have been planning to fall into a crevasse and so may not wear the correct Emergency Cold Weather gear. In this case death by hypothermia is likely. The wedge: A crevasse is generally wider at the top and narrower at the bottom. A falling human body will gather speed and become wedged between the walls of ice. When this happens it is likely the chest will be compressed limiting or preventing normal aspiration and the victim will suffocate.

Complicating factors:

It is difficult to see a crevasse from the surface of the ice. In addition, the crevasse may be covered by a snow bridge which may not bear the weight of humans or vehicles. This can make it difficult to get to the rescue site.

Noises inside a crevasse will be absorbed by the ice walls. Consequently, only sound traveling straight upward can exit the fissure. In a crevasse no one can hear you scream.

Depending on the depth of the crevasse and how tightly wedged the victim may be, a winch may be necessary to haul out the person. People have died in Antarctica, having suffocated slowly in a crevasse while the rescue team figured out how to get a truck with a winch to the site. Had they survived long enough, they would have been yanked mercilessly by a steel cable connected to a rotating drum in a torturous rescue akin to being ripped apart on the rack of the Marquis de Sade.