To claim prizes from an event players must first collect enough war credits to afford the prize.

Once you have enough war credits, navigate to the section of the prize store/rewards screen that contains the prize you would like. Once there, click on the image of the prize, then the claim button, and go through the graphics showing the prize that was claimed and how many of that prize is in inventory. If it is a blueprint for equipment, you will be given the immediate option to equip the blueprint to the unit.

Example of the starting screen of the rewards/prize store tab.

Example of what happens when there are not enough war credits to claim a prize.