If you own the game, you don't need to buy it again to get the latest version.
If you got a key that you claimed, make sure you are logged into the account that you claimed it with. If you are, there should be a download option on the store page, at the top or the bottom, in addition to the option to buy it. Alternatively you can look in your account's game library to find the download link there, alongside that of other games you may own here on Itch.