Could be bad battery cables, or possibly bad starter.

If the battery terminals were very corroded, my guess is that the corrosion also extends into the cables themselves. The corrosion is a bad electrical conductor which causes a large voltage drop to occur in the cable rather than across the starter motor. This means you're not getting enough current to the starter motor for it to turn the engine over. There is just enough voltage to turn on the starter solenoid, which is what you hear clicking in the video.