It has been mentioned that the reactions require the presence of phospholipids and calcium. The current theory is that phospholipids are provided by platelets, i.e. much of the coagulation cascade occurs on the surface of platelets. Damaged endothelial cells also contribute phospholipids. Platelets exposing suitable membranes must be triggered to adhere at sites of tissue injury. Overall, here is what happens.
Vascular injury leads to the margination of platelets to the site of vessel injury.
The platelets adhere to the damaged endothelial cells via vonWillebrand Factor . The platelets are then stimulated to degranulate and express receptors for Factors V and VIII as well as leukocytes. At this same time, blood is exposed to tissue factor and the clotting pathways are initiated. This eventually leads to the formation of fibrin. Since fibrin strands are formed in the vicinity of platelets, the two can easily come together and form a clot.