Edit: apparently the first time you claim a Steam key on Itch.io it has to verify that you have a Steam account. After this it will just directly give you the key. The interface for this verification was temporarily broken, but has been fixed by Itch.io.
On your download page on itch.io there should be a link to "Claim Steam Key". Clicking it should give you a key (a 15 digit string), this link should not take you to any other website.
You can then copy that key, open the Steam client. Go to the Games menu at the top of the window and select "Activate a Product on Steam..." where you can paste that key.
I believe you can only activate a Steam key through the client application. There is no way I know of to do it through Steam's website. I'm not sure what link would take you there, or why?
Itch.io has some official instructions here:
https://itch.io/docs/buying/already-bought
If that doesn't work, you can let me know but if there's a problem with itch.io or Steam's services I don't have a lot of control over that, we may have to contact their support. Sorry about the inconvenience.