to cool off ur balls. after an intense workout and u sit down ur balls will be on fire how much more with those boxing shorts and cup. same thing with why they put water on their heads and the back of their neck.
Sometimes to cool them down but also to wake them up with they shock of ice water. They never admit it and we don't want to know it but it also covers the need to relieve himself in the corner.
as you may know vigorous exercise like boxing means perspiration is at its highest especially in those regions so yeah to cool them down faster instead of your body diverting its own energy to cool it down ie sweating more.