The electrolyte in that clip is a solution of $\ce{NaCl}$ and acetic acid (vinegar). The zinc anode is oxidized, resulting in $\ce{Zn^2+}$ ions going into the electrolyte. The electrons went through the red LED, lighting it, and then on to the copper cathode. From there, they reduced some $\ce{H+}$ to hydrogen gas. So two $\ce{H+}$ ions were reduced to one $\ce{H2}$ molecule (which left: it's a gas) and the $\ce{Zn^2+}$ ion simply replaced the two $\ce{H+}$ ions.
Acetic acid is a weak electrolyte, so a solution of it (vinegar) is not a great conductor of electricity. The $\ce{NaCl}$ just makes the electrolyte a better electrical conductor: movable ions are necessary for a steady flow of electrical current in this battery. Without the $\ce{NaCl},$ most of the generated voltage would have been wasted on the electrolyte's resistance.
No sodium is produced (by reducing sodium ions) because the $\ce{H+}$ ions get the electrons. And no chlorine gas is produced (by oxidizing chloride ions) because the $\ce{Zn}$ anode is more readily oxidized.
