Two basic reasons.

When we cry, the fluid from the tears gets absorbed back into the tissue around our eyes (osmosis). This happens to a greater extent if the tears are because of something emotional. Emotional tears are less salty than our usual tears. The natural salt in the fluid already around our eyes basically 'attracts' the extra water into the tissue around the eyes.

In addition to all that...we get an adrenaline rush when we experience strong emotions, like sadness or anger or anything else that can make somebody cry. The adrenaline itself causes blood rush to the skin -- it literally increases swelling. That's why you feel hot when you're angry, and flushed when distraught. All of your muscles, including the muscles around your eyes, are literally filling up with extra blood. This makes the eyes look swollen, too.

The fastest way to get puffy eyes to un-swell is generally said to apply something cold; a cold ice pack, cold tea bags, cold cucumbers -- whatever you have handy.