How does aspirin "thin" blood?
Aspirine irreversibly inhibits the enzyme cyclooxygenase. This enzyme facilitates the reaction from arachidonic acid to prostaglandin G2/H2. The further reaction leads to the generation of Thromboxane A2 which is important for the activation and aggregation of platelets. See the figure (from here) for an overview:



Thromboxane A2 usually stimulates the activation of new platelets and increases their aggregation, with a shortage of this compound these activations are not possible and the coagulation of blood is reduced.