This question has many possible answers, depending on how deep you want to go into the philosophy of chemical thermodynamics.
As a simple response, K values are calculated as ratio of products over reactants. If your equation is
A + B + C => D
then the K is D/ABC
But what if A and B are really the same chemical?
A + A + C => D
then the K is D/AAC
In another words,
2A + C => D
has a K of D/A^2C
The coefficients become exponents because the K calculations involve multiplication and division, not addition and subtraction.
If you want to discuss this further, by all means ask a deeper question. For instance, why are K values based on multiplications and not additions? Or, is there a way to convert from concentration multiplications to additions of something else? I've spent a lot of time pondering these issues.