Is there a reason for the mathematical form of the equilibrium constant? Why are the two molarities multiplied and not added, and why is each raised to the power of the coefficient rather than multiplied by it? What is the reasoning behind this form? Was it simply determined that the probability of reaction was proportionate to both and thus they are multiplied together? I'd like to know whether its form is purely empirical or has a theoretical backing.
Equilibrium is also the state in which the forward and reverse reactions are proceeding at equal rate. Consider a hypothetical reaction 

$$\ce{aA + bB} \overset{k_1}{\underset{k_{-1}}{\ce{&lt;=&gt;}}}\ce{ cC + dD}$$

If we assume that the forward and reverse processes are elementary reactions (a big assumption), then:

$$rate_1=k_1 [\ce{A}]^a [\ce{B}]^b$$
$$rate_{-1}=k_{-1} [\ce{C}]^c [\ce{D}]^d$$

At equilibrium:
$$rate_1=rate_{-1}$$
$$k_1 [\ce{A}]^a [\ce{B}]^b=k_{-1} [\ce{C}]^c [\ce{D}]^d$$
$$\dfrac{k_1}{k_{-1}}=\dfrac{[\ce{C}]^c [\ce{D}]^d}{[\ce{A}]^a [\ce{B}]^b}$$

The ratio of the rate constants is the equilibrium constant:

$$K=\dfrac{k_1}{k_{-1}}=\dfrac{[\ce{C}]^c [\ce{D}]^d}{[\ce{A}]^a [\ce{B}]^b}$$