The blood dripping from a rare steak is not actually blood at all! It’s a protein called Myoglobin, an iron and oxygen binding protein found in the muscles of mammals. The protein is there to allow operation of muscles when there is a lack of oxygen from the blood supply (for example when you hold your breath) which is why marine mammals have a very high concentration of the stuff.
It’s red color comes from the iron, and when exposed to enough heat (cooked well done) the iron is oxidised becoming brown instead! When the meat is preserved in nitrates the iron bonds with them, turning a bright pink color like with corned beef or ham.