Proof of existence of lowest temperature $0 K$

In physics, temperature and other concepts in "thermodynamics" (that was known for centuries from macroscopic analyses of the heat engines and similar systems) is given by a more fundamental theory, the so-called "statistical mechanics". According to statistical mechanics, the thermal phenomena are explained by the motion of the atoms and various states in which the atoms may be found (and the number of these states).

In particular, the probability $p_k$ of a state $k$ (in classical physics, the state is described e.g. by the location and velocity of each particle) is given by $$ p_k = C \exp(-E_k/k_BT) $$ where $E_k$ is the energy of the state $k$, $k_B$ is Boltzmann's constant converting kelvins to joules, and $T$ is the absolute temperature in kelvins. The coefficient $C$ is a "normalization factor" that is $k$-independent and chosen so that the sum of $p_k$ over $k$ is equal to one (the total probability).

This form makes it clear that $T\lt 0$ isn't allowed: the exponential would be growing with $E_k$ and because there are infinitely many states with ever larger values of $E_k$ (the kinetic energy may grow arbitrarily high, in particular), the probabilities would be getting larger and their sum would diverge: it couldn't be normalized to one.

Before this statistical explanation involving Boltzmann's constant was known, the temperature was a phenomenological quantity measured by a thermometer. One was actually uncertain about any redefinition $T\to f(T)$ where $f(T)$ is a monotonically increasing function. In principle, one may relabel $T$ so that zero kelvins gets mapped to $-\infty$ in another convention for the temperature, for example; try $T_\text{new convention} = \ln (T)$. However, the ideal gases obeyed $pV = nRT$ so at a fixed pressure, the volume of some gas was proportional to the absolute temperature – the same one as one in statistical mechanics, without any redefinition by a function $f$.

So people knew how to measure the "right absolute temperature" even well before statistical mechanics was understood. The usual thermometers relied on the expansion of liquids etc. which are not ideal gases but they're close enough. For ideal gases, where the absolute temperature is proportional to the volume, the statement that $T\gt 0$ is equivalent to the statement that the volume of the ideal gas cannot be negative. You cool it down and it shrinks but it can't shrink below zero.

Volume is about the "shape" but the underlying reason for the positivity of temperature isn't about locations; it is about the motion. Any physical object with quadratic degrees of freedom will carry $k_BT/2$ of kinetic energy per degree of freedom. Again, because the quadratic kinetic energy of the type $mv_x^2/2$ can't be negative, the absolute temperature can't be negative, either.

In lasers and similar devices, one may formally find negative absolute temperatures when the number of atoms at a higher energy level is greater than the number of atoms at a lower energy level. However, this negative temperature can't be brought to equilibrium with all degrees of freedom in a larger object because the number of high-energy states is always divergent. In lasers, one kind of abuses the fact that that the energy of the "interesting degrees of freedom" is bounded both from below and from above (we only allow two or few levels for each atom).

Zero kelvin is the temperature at which there is no thermal motion. Since temperature by definition is the average thermal motion (really kinetic energy) of an ensemble of molecules, then it is a matter of definition that there can not be a lower temperature than that zero K--because there is no such thing as "negative" motion. A good analogy is a batting average. You can't have a negative batting average because there are no "negative" hits. Or, as they told me when I was an intern--you can't get less than zero sleep.