Based largely on a prior answer to the question: 'Why is it important to use a salt bridge in a voltaic cell? Can a wire be used?', the explanation centers on maintaining the charge balance (electrical neutrality) of the battery cell.
The electrons are efficiently transported via the electrodes connected by wires as sourced from the half-cell:
$\ce{Zn -> Zn(II) + 2 e-}$
However, to balance the charge in this half-cell, NO3- must enter from the salt bridge (and conversely, K+ must enter the other half-cell to balance the charge in both that cell and the salt bridge).
Not surprisingly, the less efficient part of this cell is the salt bridge itself creating resistance. Raising the concentration in the salt bridge lowers relative water content (a poor conductor and source of resistance) and increases the potential rate of diffusion of the respective NO3- and K+ ions, resulting in a corresponding increase of the cell efficiency (as witnessed by an increase in voltage). Some supporting comments per Wikipedia:
If no salt bridge were present, the solution in one half cell would accumulate negative charge and the solution in the other half cell would accumulate positive charge as the reaction proceeded, quickly preventing further reaction, and hence production of electricity.[1]
Also:
The conductivity of a glass tube bridge depends mostly on the concentration of the electrolyte solution. At concentrations below saturation, an increase in concentration increases conductivity. Beyond-saturation electrolyte content and narrow tube diameter may both lower conductivity.
And with respect to filter paper bridges, more details:
Conductivity of this kind of salt bridge depends on a number of factors: the concentration of the electrolyte solution, the texture of the filter paper and the absorbing ability of the filter paper. Generally, smoother texture and higher absorbency equates to higher conductivity.
Another interesting cited desirable property of the salt bridge is that the mass ratio of the bridge electrolyte relative to half-cell electrolyte should be small.