In the Daniell battery, the copper electrode (cathode) attracts electrons from the zinc electrode (anode). These electrons flow in the electric circuit. As soon as the copper electrode receives the electrons, the Cu²⁺ ions present in the copper sulphate solution approach to the electrode to equilibrate the negative loads. When the Cu²⁺ ions reach the surface of the copper, they react with two electrons, depositing in the electrode under the metal form. On the other hand, for each deposited copper ion, a Zn⁺⁺ is freed from the zinc electrode to the zinc sulphate solution.

As well as for any battery, the Daniell battery doesn't last forever. In fact, the production of electric current decreases as the concentration of the zinc electrolyte increases and the copper electrolyte decreases. In the particular case of the saline bridge, the function of it is to allow the transport of charges from one electrolyte to the other. As it is accumulating positive charges (Zn⁺⁺ in the zinc sulphate electrolyte, the Cl^- of the NaCl aqueous solution compensates it swinging charges. Therefore, the concentration gradients (polarization) are minimized at least for the initial instants.
The Daniell battery can produce higher potentials when several independent units are associated in serie. Don't forget to try the science home!