The items that I just bought do not appear in my backpack!

Steam might be having some connection issues with your backpack, try opening and closing TF2 (or close it if you already have it open) and check your Steamcommunity backpack.

If the items still do not show up then the trade might not have gone through, you will have to check your trade history and verify that the trade went through.

Items disappear in-game when they are used in a trade offer. This is enforced by Valve to prevent item-duping exploits, if the trade offer failed to go through, the items will be returned after a map change or you disconnect from the server you are on. We have no influence on this behavior.