Why do air embolisms kill people?

No oxygen: Air embolism will cause occlusion to artries depending on location of the artery. In lungs will cause sudden decrease in oxygenation of blood and can induce stress on the heart and death. In the brain can cause stroke.

Block Pulmonary Flow: Air embolism is caused by air bubbles in the vascular system; venous air embolism can result from the intro of air through IV lines, especially central lines, and generally must be substantial to block pulmonary blood flow and cause symptoms. Arterial air embolism in even small amounts can cause death by blockage of coronary or cerebral arteries.