What does it mean for chemical reaction network to be in equilibrium?
Ignoring the math, "in equilibrium" means the forward and reverse reactions are proceeding at the same rate, so there is no gross change in the amount of reactants. In a closed system, if initially one reaction was faster, then it's product will start to dominate, and, by the law of mass action, the reverse action will "catch up" because there is now a larger supply of those reactants.

A simple physical analog:


Balance a ruler on two fingers, one finger near the end of the ruler, the other finger inset from the end towards the middle. 
Gradually move your fingers together.


No matter where you start, the fingers will come to rest near the middle of the ruler.