The following is a very simplified version of the science but, without getting into university-level quantum physics, it will (hopefully) suffice.
The molecules in all solids are not at rest. They're vibrating. It's the energy from these vibrations which is perceived as 'temperature'. Additional energy can be added by, for example, placing the solid in a flame, which results in the greater molecular movement (i.e. a higher temperature). If enough energy is applied, the bonds between the molecules start to break down and the solid becomes a liquid. Adding yet more energy (heat) means that the molecular activity is sufficient for individual molecules to break free from their neighbours, as the liquid becomes a gas.
Going the other way, as we remove energy (heat) from any substance, the molecular activity decreases. (i.e. the temperature drops). Eventually there comes a point at which there is no molecular movement whatsoever, so the temperature is as low as it can possibly go. This temperature is defined as (absolute) zero on the Kelvin scale. (i.e. 0K)
A temperature lower than 0K would have to refer to a level of molecular activity which is less than 'not moving at all'. Clearly, that's impossible, so nothing can ever be at a temperature below 0K.
Chris