Why ? Because nature of matter at molecular and atomic level is dynamic, not static. In classical mechanics, objects can be in long term mutual rest. Not in quantum mechanics and quantum chemistry.
Molecules have zero knowledge about the system being in equilibrium or not. If the process is supported by the thermodynamic and kinetic aspects, nothing is going to stop it. It may be only counteracted by the opposite process, leading to the zero macroscopic net effect.
The molecules at given temperature do not share the same mechanical energy, but their energy follows particular statistical distribution. Even if we set all these energies the same at some moment by a magic wand, they would get redistributed again in order of nanoseconds.
For water and ice, some of molecules of ice have therefore high enough energy to leave the ice structure, while some molecules of liquid water have low enough energy to get trapped by the ice lattice. At melting point, both processes are balanced, if their is zero external heat transfer.
The dynamic bidirectional state of the dynamic equilibrium applies not only to chemical equilibriums, but also to thermal, radiative and quantum mechanical equilibriums.
E.g. 2 objects of the same temperature simultaneously provide and receive thermal energy from and to each other, with molecules a/o atoms continually exchanging kinetic energy.
Similarly, molecules of air, that seems to be in rest for us, are moving by speed of a supersonic fighter plane ( and some of them even faster ), mutually colliding in average 10 billion times per second, all the time redistributing their kinetic energy.