prize, reward, give award to
奬 centers on giving a prize or reward to recognize achievement. The idea of encouraging or recommending someone by awarding them is a natural extension.
奬 is a variant form of 獎. It combines 將 (general, leader) with 大 (big), suggesting a leader giving a big reward or encouragement. The exact historical development is uncertain, but the component relationship is clear.
A general (將) stands over a big (大) prize, ready to give an award to someone deserving.
For ショウ, imagine a general giving a big prize at a show: show -> ショウ, and the award ceremony is a grand show.