Experimental verification of the field effect
The inductive effect is transmitted through bonds; the field effect is transmitted through space.

The $\ce{C-Cl}$ bond is polarized due to the electronegativity difference between the two atoms (carbon 2.55; chlorine 3.16), the chlorine end being more negative than the carbon end. We could note this excess of electron density (partial negative charge) around the chlorines in your structures by placing a ${\delta^{-}}$ by the chlorine and a ${\delta^{+}}$ by the carbon.  

Now, when the acid ionizes there is also a negative charge associated with the carboxylate group.  When the chlorines are on the same side of the bridge as the carboxylate (syn isomer) the (through space) field effect acts to destabilize the situation because we have placed the partial negative charge around chlorine closer to the negative charge on the carboxylate (placing the negative charges closer together is destabilizing since like charges repel). The destabilization results in a higher activation energy being needed to reach the transition state for ionization.

On the other hand, the anti isomer with the chlorines further from the carboxylate will be less destabilized because the negative charges are further apart.  This means that the anti isomer will have a lower activation energy for ionization compared to the syn isomer.  Therefore, the anti isomer will be more acidic (display a lower pKa) than the syn isomer due to the field effect.