To download the game again, go to your itch.io game download page. (To go there, click the download link in the purchase email you got after purchasing the game. If you don't have that email anymore, go here to have itch.io send the purchase email again here, under 'Recover Purchases': itch.io support -- but then keep that email because you can use it again in the future to download the game again.)
Then, if you previously created a user account in the game, then click the 'Pull Files from Server' button after the game starts up. That will sync all your game data on the new computer.