Only download iOS firmwares from the direct links. As long as the download links in external sources leads to the Apple download server, you're fine. Assuming you're trying to restore to 7.0.3, hit shift (Windows) or command (OS X) and click the "Restore" button and point iTunes to the IPSW that you downloaded. That's if you want to restore to a clean copy of iOS, which wipes your device.
If you just want to update (suggested only if you're already on iOS 7.X of some kind), you can shift/command click the "Check for Update" instead of the Restore button, and it'll upgrade your iOS and your data will be retained