Step 1. Download and install AnyTrans on your computer. Step 2. Click iCloud icon, and log in with your iCloud account (your Apple ID and password). How to Access Text Messages on iCloud â Step 2. Step 3: Click the button to get full list of the iCloud content categories you can manage, and then select iCloud Backup. How to Access Text Messages on iCloud â Step 3. Step 4. Select a backup and download it, and then choose Messages.