If we don't drink water for a long period of time(say,10 hours), as during fasting and then when we break our fast by drinking water why is it that our body starts sweating?
During the time when you are not drinking water you still loose some water to sweating and urine. Sweating is (especially in hot climates) an important function of the body to maintain the body temperature in desired range. If this water is not replaced, the body gets dehydrated. This induces a reaction which reduces the activity of the "body thermostat" to tolerate a higher body temperature without excessive sweating to maintain the fluid balance of the body.
When you start drinking again after this the dehydration period, the body starts sweating to regulate the body temperature again, since now the loss of water is not critical anymore for the body. For a detailed discussion (including further references) have a look at this publication: