An antistatic wrist strap should be worn any time work is being done inside the computer. The alligator clip should be attached to a non-painted metal surface on the computer case. The antistatic wrist strap prevents static electricity from building up. Static electricity can electronically destroy many of the delicate components inside the computer. As an extra precaution, it's a good idea not to shuffle around while working inside the computer case. If you need to change position then let go of any electronic component, change your position, get comfortable again, then touch the non-painted metal surface of the computer case with your hand before touching anything else. Touching the non-painted metal surface of the computer case with your hand is another way to harmlessly release any static buildup. But this is an extra precautionary step - you should still always wear the antistatic wrist strap while working inside the case for any reason.
Good anti-static wrist straps are not expensive,
such as this Belkin anti-static wrist strap from Amazon.