Why do we sweat after drinking water and running?
Exercise, such as running, increases muscle activity.  This increases the energy demand of these tissues, which increases the rate of cellular respiration.  Respiration releases heat as a by-product, therefore the body is hotter during and after exercise. 

Sweating is a homoeostatic mechanism to keep core body temperature constant. It is a response to lower the body temperature. When the body becomes too hot, sweat is released onto the surface of the skin.  The water from the sweat then takes some of the excess heat energy from the body and uses it to evaporate.  Because water has a relatively large specific heat capacity a lot of heat can be carried away by this method.