Do the units of the law of mass actions convey any meaning?
Well, you can calculate difference in number of moles of gases and active liquid species on reactant and product side species using units of $K_c$ and $K_p$, respectively, as these are mostly used for homogeneous reactions.

For example, the unit of $K_c$ is $\text{(concentration)}^n$ if $n &gt; 0$, more liquid or aqueous (ionic) species on product side.

Then, you can predict whether the product is formed or reactant when you increase or decrease external pressure or volume of a container containing a mixture in equilibrium. Temperature's effect would be only heat of reaction dependent.

Hint: write $K_c$ in terms of moles and volume to check effect of volume, write $K_p$ in terms of mole fractions and total pressure to check effect of external pressure.