God's Hand is shown through man's hands. When God wants to touch the material universe He does it through man. If He does it any other way it's called a miracle.
How were the people going to know it was God's hand, Moses asks. God responds:
First, God asked Moses what was in his hand. When we give God our hands He will show us His hand.
Second, God shows Moses what He has in store for the Israelites - complete restoration (like a leprous hand into a healthy hand).
Two ways God worked through Moses:
1) With his hand grabbing hold of his staff - when he struck the Nile, struck the Red Sea, struck the rock, etc.
To us, the staff represented the cross. We must hold onto our cross and carry it just as Moses carried his staff.
To them, the staff represented the Hand of God.
2) With just his hands - when he held them up and prayed, when the army defeated Amalek, etc.
Both of these were how God told Moses He would reveal Himself as the "I AM" to the people.
Moses used the staff for the first three, only his hands for the remaining, and no hands for either plague of death.
Why the hands? Because with the hands we WORK!

"Whatever your hand finds to do, do it with all your might." It's God Who's doing it - you're just using your hands!