Why does an electrochemical cell work?
Maybe you would understand better with an image, or if you admit that all metals &quot;are wanting&quot; to loose electrons. And in order to do that, they must find another atom which is weak enough or &quot;humble&quot; enough to accept to work &quot;against its will&quot;. In other words, some atoms are more &quot;willing&quot; than others. It is like a fight, with a &quot;strong&quot; and a &quot;weak&quot; atom. In your copper-silver cell, copper is more assertive : it imposes its electrons to the &quot;weak&quot; silver atom. Silver cannot emit its own electrons. It is even obliged to accept electrons from copper and have these electrons react with its proper ions. producing more silver atoms. Some atoms are stronger than copper. Zinc for example, opposed to copper, imposed its electrons to copper. This order of &quot;strength&quot; is experimental, and cannot be derived by calculations.
Now to speak more scientifically, you may replace the present notion of strength by the reduction potential. The more the reduction potential is negative, the more the metal has a tendency to loose its electrons.
Anyway, a salt bridge is absolutely necessary for the cell to work. If there is no salt bridge, it is not a cell, and it produces no chemical reaction, and no electric currant.

Complement, required by Manit.
If you want a local phenomena to explain the cell, you may consider the following development. You may admit that copper &quot;wants&quot; to create more $\ce{Cu^{2+}}$ ions in solution. These ions are positively charged. So the copper compartment must attract negative ions to compensate the new positive charges appearing around the copper plate. As a consequence, the new $\ce{Cu^{2+}}$ ions attract negative ions from where they are available, namely in the other compartment. And as $\ce{Ag}$ is not able to create $\ce{Ag+}$  ions with the same force, Silver plate must admit that it looses its negative nitrate ions to send them to the copper compartment. And as a consequence, it must admit that the positive ions $\ce{Ag+}$ should also &quot;disappear&quot; from the compartment. And the only way to make this charge disappear from the compartment is to accept electrons, so that the $\ce{Ag+}$ ion is transformed into metallic silver.
OK ?