You can automatically import the Objekts you own by providing your wallet address, but don't worry, the Objekts never leave your account, they can only be transferred via the official Cosmo app, this site just reads and inputs them like you would do so manually.
To find your wallet address in the Cosmo app tap the “Activity” tab at the bottom right of the screen, then click the octopus picture at the top right, and finally, you can go ahead and tap the “Copy Address” button.