The ice and water are not at equilibrium because the water is absorbing heat from the surroundings.
If the ice and the water were in an insulated container, they would be at equilibrium. The rate at which water molecules leave the surface of the ice would equal the rate at which they return to the surface of the ice.
Your glass of ice and water is not insulated. It is absorbing heat from the hot surroundings. The water transmits this energy to the surface of the ice.
More water molecules now have enough energy to leave the surface of the ice. They leave the surface faster than water molecules return from the liquid. The ice starts to melt. We no longer have a dynamic equilibrium.
As long as some ice is present, the temperature of the water will stay at 0 °C.
Once the last bit office has melted, the temperature of the water will start to increase. There will be no ice to absorb the added heat.