Decrease in molar conductivity with increase in concentration
The equation you quote is the empirical relationship discovered by Kohlrausch (1907) for the conductivity of strong electrolytes, and is valid for concentrations up to $\sqrt{c} \approx 0.3$. $\Lambda_m$ is the molar conductivity usually given with units $\mathrm{\Omega^{-1}\,cm^2\,mol^{-1}}$ ($\Omega$ = ohm).  The $\Lambda_m^0$ is the molar conductivity at infinite dilution which effectively means that of pure water.

The reason that the conductivity falls with concentration is due to inter-ionic forces on the ion mobilities. There are two related effects, the Electrophoretic and Relaxation Field effects.

Each ion is surrounded by a 'cloud' of oppositely charged ions and when a field is applied there is simultaneous movement of an ion in one direction and of the opposite movement of the ionic atmosphere; the electrophoretic effect. Both the central ion and its atmosphere drag solvent molecules along with them  and this results in a retardation of the movement of the central ion. The effect can be evaluated using Stoke's Law $v=f/6\pi\eta L$, where velocity is $v$ and force $f=zeE$ for an ion of charge $z$ in field $E$. The solution viscosity is $\eta$ and $L$ the Debye length which is the effective radius of the ionic atmosphere, typically a few nm in millimolar solution. The velocity is identified with the deceleration of the ion caused by the ionic atmosphere moving in the opposite direction. 

The second effect is a relaxation effect. Suppose the central ion is randomly moved, the overall spherical symmetry that existed before any movement occurred will be broken, i.e. the central ion is asymmetrically located with respect to its atmosphere, and because of this a 'relaxation field' is set up. The symmetry is restored by the effect of the relaxation field which takes only ten to hundreds of nanoseconds.  

When the ion is under the influence of the applied field the time average of the forces on the ion are therefore no longer zero. The external field driving the ions through solution is reduced by the continual effect of the relaxation field. This seems strange since this field relaxes so very rapidly compared to motion of the central ion through its atmosphere, $\approx 10^{-3}$ sec, however, its magnitude is huge $10^5$ V/cm compared to the applied field and even small changes in symmetry do have a measurable effect in acting against the external field.