The equilibrium constant K is just the ratio of the reaction rate constants: 
    [ G ] [ B ]     k_f
K = ----------- =  ----
    [ R ] [ Y ]     k_r
That is, with the default conditions of k_r = k_f = 0.250 the equilibrium constant K = 1. If you start off with equal numbers of red and yellow, and of green and blue molecules, so that [ R ] = [ Y ] and [ G ] = [ B ] at all times, then at equilibrium you will have the same number of all four types of molecule, because the only way in which 
    [ G ] [ B ]    
K = ----------- =  1
    [ R ] [ Y ]
is if [ R ] = [ G ] = [ B ] = [ Y ]. If, on the other hand, you set k_r = 0.750 and k_f = 0.250, then K = 1/3, and at equilibrium you'd have 1/3 as many green or blue molecules as red or yellow molecules, again assuming you started off with [ R ] = [ Y ] and [ G ] = [ B ], so that there are enough partners for every molecule to have a chance to react.

But why?

So where does the very first expression above come from? Well, we begin with the observation that what we mean by equilibrium is that the concentrations of the reactants are no longer changing. That means the rate of change of the concentrations must be zero. Mathematically we write this statement as: 
  .      
[ R ] = 0
  . 
[ Y ] = 0
  .
[ G ] = 0 
  .
[ B ] = 0
Where the expressions on the left-hand side (with the dots on top of the [  ]) mean "the rate of change of [ R ]", "the rate of change of [ Y ]", and so on.

What determines the rate of change of [ R ] ? It must be given by the rate at which G and B molecules are colliding and turning into R and Y molecules minus the rate at which R and Y molecules are colliding and turning back into G and B molecules, 

  .
[ R ] =  (rate of G + B --> R + Y) - (rate of R + Y --> G + B)

Now what is (rate of G + B --> R + Y) ? It must be 

k_r [ GB ]
where [ GB ] is the equilibrium concentration of "colliding" pairs of G and B molecules. That is, the rate of reaction is equal to the concentration of colliding molecules ( [ GB ] ) times the probability that each collision leads to a reaction (k_r).

Well we know k_r, because we chose it. But what is [ GB ] ? We can estimate it as: 

[ GB ] = [ G ] P_GB
where P_GB is the probability that a B molecule is right next to any particular G molecule. That is, the concentration of colliding pairs ( [ GB ] ) is equal to the concentration of G molecules ( [ G ] ) times the probability that a B molecule is next to any given G molecule (P_GB).

There are ways to calculate P_GB carefully, but a crude estimate is just 

P_GB = [ B ]
That is, we could assume the probability of finding a B molecule next to a G molecule is the same as the average probability of finding a B molecule at any point in the box. This assumes that the presence of the G molecule doesn't change the probability of finding B molecules in the neighborhood. That is clearly not exactly true, but it's not a bad start.

Now we can assemble all our pieces: 

P_GB = [ B ]
so 
[ GB ] = [ G ] P_GB = [ G ] [ B ]
and hence 
(rate of G + B --> R + Y) = k_r [ GB ] = k_r [ G ] [ B ]
and following the same logic 
(rate of R + Y --> G + B) = k_f [ RY ] = k_f [ R ] [ Y ]
so that finally 
  .
[ R ] =  (rate of G + B --> R + Y) - (rate of R + Y --> G + B)
becomes 
  .
[ R ] =  k_r [ G ] [ B ] - k_f [ R ] [ Y ]

Now remember our initial English statement ``we have reached equilibrium'' is equivalent to the mathematical statement: 

  .
[ R ] =  0
Using this fact and our new equation we get: 
  .
[ R ] =  0 = k_r [ G ] [ B ] - k_f [ R ] [ Y ]
Which can only be true if 
k_r [ G ] [ B ] = k_f [ R ] [ Y ]
Rearranging this equation we find: 
[ G ] [ B ]     k_f
----------- =  ----
[ R ] [ Y ]     k_r
Which is exactly what we asserted at the very beginning. Science works!
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