Please explain the mystery of the equilibrium constant expression for chemical reactions
I think the most intuitive way to think about this is to view it through a kinetic lens. That is, recall that at equilibrium, the rate of the forward reaction and the rate of the reverse reaction must be equal. When we write expressions for the rates of reactions, we multiply the reactants because the probability of collision is a function of the product of the two concentrations, not the sum. This only really makes sense for an elementary step, but the result applies to the whole reaction.

Let's consider the simpler reaction $\ce{A + B &lt;=&gt; C}$.

If there's is only one step in the reaction, the rate of the forward reaction will be given by $k_\mathrm{fwd}[A][B]$ and the rate of the reverse reaction by $k_\mathrm{rev}[C]$. 

At equilibrium, these are equal, so 

$$k_\mathrm{fwd}[\ce{A}][\ce{B}] = k_\mathrm{rev}[\ce{C}].$$

Rearranging,

$$\frac{k_\mathrm{fwd}}{k_\mathrm{rev}}=\frac{[\ce{C}]}{[\ce{A}][\ce{B}]},$$

and we see the product of the concentrations in the equilibrium constant.

You can also go the route of deriving the expression from the free energies and Boltzmann distribution, but I don't think that's nearly as intuitive as thinking in kinetic terms.