Why the salts in a salt bridge?
The purpose of the salt bridge is to prevent the two half-cell solutions from mixing. It is possible to make a really bad galvanic cell by putting both half-cells in a single solution, but they rapidly self discharge as the oxidizing agent ($\ce{Cu^+}$ in your example) can diffuse through the solution and react directly with the other electrode (Zn in the example), with the electrons being transferred directly instead of going through the external circuit.

Having a "bridge" of just water soaked paper is similar to having both half-cells in the same container but poses some issues. First, the diffusion of ions through the bridge is going to be very slow, increasing the cell resistance, second, the self-discharge problem above will occur, and third, because different ions diffuse at different rates, a junction potential develops as the two containers diffuse into one another.

So a salt bridge's job is to not only maintain electroneutrality between the two containers, but to do so without allowing the solutions to mix. An inert salt is used to prevent any reactions from occurring between the salt and the solutions, and things like $\ce{KCl}$ are commonly used for analytical cells as potassium and chloride ions have very similar diffusion coefficients, minimizing junction potential.