The experiment is a simple break down on how common voltaic battery works. A battery usually consists of 2 important parts known as the electrodes and the electrolytes.
We also need to understand the basic structure of an atom because they are components that carry positive and negative charges. By law of nature, an atom carries an amount of these charges. Since we are all made of atoms then we all carry a charge. Moreover, positive and negative charges are attracted to each other. By default, our atoms are in neutral state because they carry an equal amount of these charges. During chemical reactions, atoms becomes unstable where they carry unequal amount of charges. In this case, an atom is called an "anion" if it carries more negative charges and it is called a "cation" if it carries more positive charge.
In a battery there are two electrodes known as the cathode and anode. Anions are drawn more towards to the anode and cations are drawn more towards the cathode. Between the electrodes is an electrolyte which is a substance that allows charged atoms (eg. cations/anions) to pass through. In the above experiment, the "Zinc" metal is the anode, the "Copper" metal is the cathode, and the "Citric Juice" is the electrolyte.
When the zinc touches the citric acid, a chemical reaction occurs called "oxidation". Each of zinc atom releases its negative charges. Negative charges cannot pass through electrolytes so it will pass through the external wire. Since the atom lost negative charge and only have positive charge left, it will pass through the electrolytes and drawn to the cathode which is made of copper.
The negative charges that traveled through the wire will be received to the copper where it gains negative charges along with the positive charges it has and results in a chemical reaction called "reduction". The result of this chemical reaction is hydrogen gas.