If the first law of newton says that for producing motion you need an external unbalanced force to be applied or in simple words we can't move by ourselves so how are we able to move our hands?

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Solution

Consider your hand as a system and everything other as external. Your hand will move if someone pushes it and that push is an unbalanced external force.
But if you want to move your hand by your own, it's not just your hand but it's the muscles inside your hand do the work for you to move your hand. That work done by the muscle is an external unbalanced force because as we've said earlier , in this case anything other than your hand is external. So it too works on the principles of Newton's first law of motion.

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