Electrochemistry molar conductivity Molar conductivity increases with decrease in concentration. This is because the total volume, $V$, of solution containing one mole of electrolyte also increases. It has been found that decrease in $\kappa$ on dilution of a solution is more than compensated by increase in its volume. Physically, it means that at a given concentration, $\Lambda_m$ can be defined as the conductance of the electrolytic solution kept between the electrodes of a conductivity cell at unit distance but having area of cross section large enough to accommodate sufficient volume of solution that contains one mole of the electrolyte. When concentration
  approaches zero, the molar conductivity is known as limiting molar conductivity and is represented by the symbol $E^\circ_m$. The variation in $\Lambda_m$ with concentration is different (Fig. 3.6) for strong and weak electrolytes.


Can anyone explain what does this sentence mean : "it has been found that decrease in k on dilution of a solution is more than compensated by increase in its volume"

Molar conductivity : $\kappa \times V$ ($\kappa$ is specific conductance), $V$ is volume of solution containing 1 mole of electrolyte.

I understand that molar conductivity increase with decrease in concentration. But at the same time this sentence is contradicting that because it says the decrease in "$\kappa$"(specific conductance) is more than increase in volume Which should decrease molar conductivity. I am confused.
I think you missed the word "compensated".


  The decrease in specific conductivity is more than compensated by increase in its volume.


This simply answers your question. The reduction in molar conductivity due to the decrease in specific conductivity is  overridden by the increase in volume.

It is saying that the latter overrides the former, and not just overrides, but overrides by a lot.