Click one of the three rocket part names (Capsule, Fuel Tank, or Engine), then click one of the upgrade buttons on the left then click the big green "Upgrade!" button. Also, you need to have the materials (you can see how much material you have on the top left or the number after the small grey box) to upgrade the current rocket part (the cost is above the "Upgrade!" button).
Hope this helps and thanks for playing :)