Why do my hands turn red when I place them below my heart?

A lot of things. Mostly circulatory reasons. It depends upon what your circulation is when they aren't down. But also could be related to other factors such as exposure to cold. You need to get checked.Out. There are also other conditions such as auto-immune disease they may be working here.
Physiological. This may be a normal physiological process in your body with increased blood flow as your place your arms down and venous return flow will be slower and may be pooling. Your physician can help clarify if you have a vascular problem.