We all know that the atoms in any solid matter vibrate about their mean position but why do they do so?? What will happen if they stop vibrating??
I will interpret your question to mean: why must all particles vibrate? This is explained best using quantum mechanics. One of the postulates of quantum mechanics is the Heisenberg Uncertainty Principle. It means that one cannot measure both the position and momentum of a particle to arbitrary precision. This means that if one measures the momentum of a particle to greater levels of precision, the position of the particle becomes less certain and vice versa. The particle must vibrate since if it was perfectly stationary, it would be possible to determine position and momentum to greater levels possible than that allowed by the Heisenberg Uncertainty Principle.