In the shop, you click on the "purchase" button under the item you want. Then it will be added to your cart. Then you'll need to click on "view cart/checkout" and finalize the purchase. Then the item will be added to your inventory.
So, [not an expert], on the right, it says view cart/checkout, it will say what you put in your cart and then pick checkout. Or something like that. :]
are you talking about buying things from the shop tab in the tbt banner? after you add something to your cart you have to hit "view cart / checkout" in your cart in the top right