Why is human blood blue in the body and red when it comes out?

Actually, it only appears blue in the veins. It is really a dark, deep almost purplish red in the veins and a less deep red in the arteries. When you cut yourself and the blood is exposed to air, it becomes bright red (as the fight announcer says, "the claret is really starting to flow now!")
One of the proteins in the blood, hemoglobin, carries oxygen to the body. When it is carrying oxygen, it changes shape. Due to its different shape, it absorbs different wavelengths of light, changing its color.
Oxygen poor blood is darker than Oxygen rich blood.

deep tread is right,
if you drew blood into a sealed tube fromyour veins, it may look a purple red color, but it would still be red.
None of the blood in your body is truly blue. It only looks blue sometimes, under your skin -- due to an optical effect involving the way pale skin, and the tissue beneath, absorb and reflect light.
blue light penetrates your skin the best, so that is what reflects off of your veins and that is what you see, but if you saw them in plain light they would look red.

the other answers are partially right: blood is not blue in the body (unless the person is severely anoxic (without oxygen)). However, it's like this: in the veins, which carry blood from tissues back to the heart and then the lungs, blood is purplish red, because of changes in hemoglobin when the oxygen is low. In the arteries, which carry blood from the lungs to the tissues, it's red. But what's not true is the concept that when cut, blood turns red as it is exposed to the air. In fact, the color of blood as seen through injury depends on what's injured: a cut vein will produce the purplish blood, and a cut artery will produce red blood. In surgery, it's easy to tell whether bleeding is venous or arterial from its color. Exposure to air doesn't significantly change the color of blood; it needs to be mixed with air on a microscopic level as in the lungs for that to happen. Most cuts bleed red because the capillaries are what bleed, and the bleeding comes from forward-flowing pressure, the parts still carrying oxygenated blood