Most batteries use two different materials and an electrolyte solution to create an imbalance of charge and thus a voltage. When the terminals of the battery are connected with a wire, this voltage produces a current.
In this Snack, the thin film of sweat on your hands acts like an electrolyte solution and reacts with the copper and aluminum plates. When you touch the copper plate, the copper gives up negatively charged electrons and acquires a positive charge. When you touch the aluminum plate, it takes electrons and becomes negatively charged.
This difference in charge between the two plates creates a flow of electrical charge, or electrical current. Because electrons can move freely through metals, the excess electrons on the aluminum plate flow through the meter on their way to the copper plate. In addition, negative electrons move through your body from the hand touching the copper to the hand touching the aluminum. As long as the reactions continue, the charges will continue to flow and the meter will show a small current.
Your body resists the flow of current. Most of this resistance is in your skin. By wetting your skin you can decrease your resistance and increase the current through the meter. Since two people holding hands have more resistance than one person, the flow of current should be less. If you like, you can use the multimeter to measure resistance directly in these situations.