Why does a dynamic equilibrium exist?
To understand the confusion here you need to take two different points of view about what is happening in a reaction: the molecular view and the statistical view.

The statistical view ("Gibbs free energy is a minimum") is an average of all the molecular activity in the reaction. It describes the net, bulk behaviour of the reaction. At equilibrium, there is no net direction of change. But this tell us little about what is happening to the individual molecules or atoms in the reaction.

The molecular (or atomic) view is about what is happening with the individual things that make up the reaction. At any temperature above absolute zero there will be plenty of things happening at the molecular level. There is plenty of energy around and the molecules will bang into each other a lot. Moreover, not all the components will have the same energy (some molecules are slow moving, others fast: the distribution of energies will follow a Boltzmann distribution). Sometimes there is enough energy for the products of the reaction to fall apart; sometimes there is enough energy for the reactions to react to give the product. Equilibrium is reached when, at a given temperature, the rates of both reactions are the same not when everything stops at the molecular level. Only when we look at the overall averages in the bulk reaction do we see no net change: at the molecular level things are always happening.

It is worth noting that the molecular view gives more insight than the purely thermodynamic view which only considered bulk averages. Some reactions at the molecular level happen fast, others slowly. Often in chemistry it is the kinetics that determine the product not the thermodynamics and thinking about the molecular point of view often explains the observed situation in the bulk. 

It is quite important and insightful in chemistry to be able to switch from one view to the other as you will get a much richer interpretation of what happens in real reactions if you can.