In a saturated solution, why is there ongoing dissolution and crystallisation?
Because equilibrium is dynamic not static

There are many things in the world of chemistry which could be simpler but are not. Equilibrium is one of them.

There are few chemical processes where equilibrium is equivalent to the situation where everything just stops. Equilibrium normally occurs when the rates of processes going forward are the same as the rates going backward, not when things stop happening. In this case the rate of dissolution is the same as the rate of crystallisation. 

On a molecular level things are very dynamic. Molecules in the solution are hitting crystals all the time. Molecules in the crystal are vibrating and being hit by molecules of the solvent: sometimes the molecules in the crystal have enough energy to escape from the crystal and go into solution. Equilibrium is the point where those two reactions have equal rates, not where all the activity stops.

We know this is true in real solutions from several experiments. We can, for example, label the molecules in the crystal with radioactive versions of one of their atoms (eg carbon 14). Even if we start with a saturated non-radioactive solution and add radioactive crystals of the same compound, the radioactivity will spread to the solution (see this question Proof of Dynamic Nature of Equilibrium). 

The key thing is that all equilibria in chemistry are dynamic. The major difference in many cases is how dynamic. Some happen very slowly, others very quickly. But the position of the equilibrium only has a weak relationship to the speed of the equilibrium.