What is the organ system responsible for taking oxygen from the outside air and supplying it to the blood into the body?

The answer should be the respiratory system.

When we breathe, air is taken in our body and they travels to the lungs through the trachea. The lungs is the site where the oxygen from the air we just breathed is exchanged with the carbon dioxide (waste).

In the lungs, there's something call air sac, which diffuses the oxygen to the capillaries. At the same time, carbon dioxide from the capillaries diffuses back to the air sac. Then, we breathe out and the carbon dioxide and the remaining substances in the air that we don't need is breathe out.

That explains why there's more oxygen in the unbreathed air than oxygen from breathed air, and less carbon dioxide in unbreathed air than breathed air.

And after that, the capillaries transfer these oxygenated blood to the pulmonary vein, and then transfer them back to the heart.