It may be useful to consider why molecules can't stay still. This phenomena is somewhat tricky to understand because it arises fully from quantum mechanics.
First, consider the Heisenberg Uncertainty Principle. This principle states that we cannot know the exact momentum and position of a particle simultaneously. If we know the exact position of a particle at a certain time, then we cannot know its precise momentum (i.e. speed and direction) and vice versa.
So, consider an unmoving particle. Since it is not moving, we know that it's momentum must be zero. Furthermore, because it is not moving, we would also know it's precise location. Therefore, an unmoving particle violates the Heisenberg Uncertainty principle and cannot exist even at absolute zero. This requirement for particles to have momentum (and therefore kinetic energy) at absolute zero is what is known as the infamous zero point energy.