The procedure works like this: a numbing agent is applied to the face then a couple vials of blood are drawn from the patient. The blood is spun in a centrifuge to separate the red blood cells from the platelets. This takes about 10 minutes. A special piece of equipment is used to puncture the skin on the face while placing the platelets on the skin's surface. The platelets are supposedly absorbed into the skin during the process. (Popular Science compares it to the procedure used when making a Jell-O Poke cake.) This Miami News site has a video that shows you how it is done.
The idea is that the platelets are absorbed by the skin and since the platelets contain growth factors, it is possible that they might cause new collagen to be formed. The collagen will plump up the skin causing wrinkles to disappear or be less visible. This is all a theory, however, as there is no real proof that it works this way.
Dermatologists say that there really is little evidence that this process will make you look younger but they also say that there is probably no harm in undergoing the procedure because a patient's own blood is used. Of course any time that the skin is punctured, there is a risk of infection but as long as the equipment is properly sterilized, this risk is reduced.
Still, it is not a cheap procedure. It can cost $1000 to $1500 per treatment. That is a little out of the range for most people for a procedure that might not actually work for you. Any results are said to be temporary - lasting maybe 6 to 8 months.
Frankly, unless you have money to burn I wouldn't bother with this treatment. It certainly is not a miracle. Besides, who ever said there is anything wrong with looking your age?
Image of red blood cells found here