Platelets, also called thrombocytes (thromb- + -cyte, blood clot cell), are a component of blood whose function (along with the coagulation factors) is to stop bleeding by clumping and clotting blood vessel injuries.
Milk of the poppy is essentially the Westeros version of morphine. Maesters make it from crushed poppy flowers, and morphine and other opiate-based drugs are made from poppy plants. Indeed, the books' descriptions strongly suggest that the poppy is intended to be the real-world opium poppy (Papaver somniferum).
The first was that it goes out of its way to state that opium poppy seeds are, in fact, legal, presumably because of their legitimate culinary uses. There seems to be a chicken-and-egg paradox here, however, in which illegal poppy plants produce legal poppy seeds from which grow illegal poppy plants.