Why is chicken meat white and beef red?

The quick n' dirty here. Basically, it comes down to myoglobin,

Red, or dark meat is made up of slow-twitch [muscles]. These muscles are used for extended periods of activity, such as standing or walking, and need a consistent energy source. The protein myoglobin stores oxygen in muscle cells, which use oxygen to extract the energy needed for constant activity.

White meat is made up of fast-twitch [muscles]. These muscles are used for quick bursts of activity, such as fleeing from danger. These muscles get energy from glycogen, which is also stored in the muscles.

In dark meat, myoglobin helps us determine when cooked meat is done

When dark meat is cooked, myoglobin's color changes depending on what the meat's interior temperature is. Rare beef is cooked to 140° F, and myoglobin's red color remains unchanged. Above 140° F, myoglobin loses its ability to bind oxygen, and the iron atom at the center of its molecular structure loses an electron. This process forms a tan-colored compound called hemichrome, which gives medium-done meat its color. When the interior of the meat reaches 170° F, hemichrome levels rise, and the myoglobin becomes metmyoglobin, which gives well-done meat its brown-gray shade.

In white meat, again it is protein that helps us tell if it is done, but since there is little myoglobin, we use another mechanism to tell if it's done

White meat has a translucent "glassy" quality when it is raw. When it's cooked, the proteins denature and recombine, or coagulate, and the meat becomes opaque and whitish.