How does an ice cube melts?

An ice cube melts basically because of the energy given by heat transforming the solid (ice) to a liquid (water) and even a gas (steam).

The ice cube comes first in a solid form and the particles in the solid are vibrating all the time, so the heat (the increasing of temperature) will give energy to the particles which will cause them to vibrate more. The particles will start to vibrate and often bump over each other, which eventually will turn out that the ice cube will turn from it solid state to it's liquid state. As the temperature of the ice increases, the particles which previously maintained a rigid formation, will begin to gain energy and begin to move The particles vibrating will be rubbing against each other, causing some sort of friction, which will make them hotter.  The freezing temperature will start to become hotter an the density between particles will decrease. When the particles are hot enough and the density is getting lower and lower, the particles will change from solid to liquid, liquid particles have more kinetic energy than solids (they move more). The solid bonds, no longer exists and now, the ice is liquid, if you add even more temperature the water will gain more energy. Eventually became a gas because the particle gained enough energy to "escape" from the liquid form, this is called sublimation.