This answer is the same as the one by Philipp, but instead of using a specific example, it makes the argument for any multi-step reaction where each step is an elementary reaction.
The overall reaction is a sum of elementary steps
The overall reaction happens because elementary steps happen consecutively. Along these multiple steps, some species are reactants or products (i.e. appear in the chemical equation of the overall reaction) and others are intermediates (i.e. made by one elementary reaction and consumed by another).
The overall chemical equation is equal to the sum of the elementary steps, with intermediates appearing both on the reactant and the product side (those can be removed to arrive at the net chemical equation of the overall reaction).
At equilibrium, all elementary steps are at equilibrium
The only way the reactant and product concentrations can be constant (i.e. at equilibrium) is when all the intermediates are at equilibrium as well. You arrive at the equilibrium constant of the overall reaction by multiplying the equilibrium constants of the elementary reactions.
The rate laws of the elementary steps have the stoichiometric coefficients as exponents
This is the definition of an elementary step. The rate law for the forward reaction is the product of the reactant concentrations raised to the power of their coefficients. The rate law for the reverse reaction is the product of the product concentrations raised to the power of their coefficients. (Many of these reactants and products are intermediates with respect to the overall reaction.) At equilibrium, the forward and reverse rates are equal, so product concentrations divided by reactant concentrations (raised to respective coefficients) for the elementary steps will equal the quotient of reverse and forward rate constants, and we will call this equilibrium constant.
The equilibrium constant expression of the overall reaction is the product of the equilibrium constant expression of the elementary steps
Because the equilibrium constant of the overall reaction is equal to the product of the equilibrium constants of all the elementary steps, we can derive the equilibrium constant expression of the overall reaction by writing the product of the individual equilibrium constant expressions.
Just like intermediates cancel out of the net chemical equation because they appear both as reactant and as product, concentrations of intermediates will cancel out of this derived equilibrium expression (they appear both in the denominator - as reactant of one step - and in the numerator - as product of a different step). What remains after canceling are concentrations of reactants and products of the overall reaction, raised to their respective coefficients in the net reaction.
So whatever the coefficients are in the chemical equation of the net overall reaction will appear as exponents in the equilibrium constant expression for that same net overall reaction.
For an example, look at the answer by Philipp.