According to the British Journal of Anaesthesia, heart rate rises during exercise in order to increase cardiac output, compensating for increased blood flow demand to working muscles. Blood flow to the muscles is increased because of the increased workload, nutrient demand and metabolic demands of the body.
According to NursingLink, exercise causes higher production of waste metabolites and heat that need to be discarded, and the British Journal of Anaesthesia further explains that blood flow in the skin is also increased during exercise for adequate heat dissipation.
NursingLink also notes that an increased heart rate from exercising is directly related to the workload increase and exercise intensity. However, as the British Journal of Anaesthesia states, the maximal cardiac output in trained athletes is achieved through a greater stroke volume rather than an increased heart rate.