Here's what the Harvard Doctor had to say in the link that Mary posted above.
What you are describing is common. It’s likely related to vasodilation, which means that blood vessels are opening up, leading to increased blood flow near the surface of the skin. Increased blood flow just under the skin makes the face pink or red and it may cause puffiness of the hands.
Contrary to popular belief, getting red in the face during exercise does not mean you are out of shape.
Vasodilation may occur as one of the body’s ways of regulating temperature. Releasing heat generated by exercise is important, as it allows the body to maintain normal temperature.
Even before working up a sweat (another means of releasing heat), the blood vessels near the skin tend to dilate. This allows more heat to be released from the blood as body temperature rises during exercise.
There may be other explanations for puffy hands during or after exercise:
An exaggerated, normal reflex. During exercise, blood flow to exercising muscles increases. In an effort to maintain circulation to other parts of the body (including the hands), blood vessels may dilate. In some people this reflex is more pronounced than others.
Trapped fluid. During exercise, there is increased circulation throughout the body. While the legs have large muscles that work hard during many exercises and that help “pump” this added blood flow through the veins, the arms have smaller muscles and they may not be working as hard during a workout (for example, while jogging). If the hands are held below the level of the heart, the added fluid must travel against gravity. Together, these factors make it harder for the fluid to return to the heart. The extra fluid that is “trapped” in the hands can cause puffiness.
Once you cool down and the circulation and blood vessels return to normal, your hands and the color of your face should return to normal. If your rings are tight after exercise, it’s probably a good idea to take them off before you start working out.
Your hand swelling and facial redness may be annoying but they are unlikely to be due to anything serious. Think of them as small prices to pay to stay in shape and avoid overheating.