Go to the top of the page and you'll see a banner that says "You own this game" - click the nearby Download button. If you're using windows, download "That's Not My Neighbour.exe" (not version 1.0.3.3!), and drag it into the folder where your previous version is - windows will ask if you'd like to replace it, say yes.
I have another comment if you scroll down, it includes pictures which may help.
If that doesn't work, look through your Itchio library to see if the purchase was actually connected to your account.