Blood clotting in mammals is done by platelets, which float around in the blood. Platelets are kind of like (simple) cells in a way. Platelets have a membrane. Inside the membrane are the molecular tools—that’s what I usually call enzymes despite being an enzymologist—they use to clot blood.

Inside platelets, chemical reactions are sped along by the enzymes. One of the important chemical reactions for clotting is the conversion of arachidonic acid (through a couple of intermediates) to thromboxane. Now, arachidonic acid is so simple: a linear, 20-carbon chain with a carboxylic acid group at one end, and four double bonds in the chain. Converting it to thromboxane takes a couple of steps: the enzymes need to kink the chain and oxidize specific carbons to make them polar.

The trouble is, aspirin inhibits the enzymes! So the enzymes inside the platelets can’t make their thromboxane. And, as such, clotting is inhibited, which results in bleeding. So this is why aspirin can cause bleeding