We take the gas in a box. The molecules collide and nothing more: no attraction, no repulsion. The energy of one molecule is then kinetic energy alone, \dfrac{1}{2}m(v_x^2 + v_y^2 + v_z^2).
Write the kinetic energy of one component as \varepsilon_x = \dfrac{1}{2}mv_x^2. Put into the Boltzmann factor, the probability becomes a product of terms like e^{-\beta\varepsilon_x}. A sum in the exponent splits into a product, so the three components are independent and each carries the same shape of distribution.
The figure shows one component. A bell centred on 0, spreading as far right as left. That being at rest is the most likely thing sounds odd, but this is one component of the velocity. The speed changes shape in the next scene.
There are many molecules with the same speed v differing only in direction. In velocity space they may be anywhere on a sphere of radius v, and the area of that sphere goes as v^2.
So the distribution of speed is the bell multiplied by v^2. The faint curve in the figure is the bell, the nearly straight rising one is v^2, and the bright one is their product. This is the Maxwell distribution.
Thanks to v^2 the region at v = 0 is pushed down. The most likely speed is no longer 0 but the top of the peak. That is what is meant by saying there are hardly any molecules at rest.
Raising the temperature stretches the tail of the Boltzmann factor and fast molecules appear. The top of the peak moves to the right.
At the same time the peak comes down. The number of molecules does not change, so the area under the curve must always be the same. Whatever it gains sideways it gives up in height.
The three faint curves in the figure are the two ends of the temperature range and the middle, and the bright one is the current temperature. A cold gas has a narrow tall peak over to the left, a hot one a low wide peak trailing far to the right.
With the distribution in hand we take the average. Each component gives \dfrac{1}{2}kT, and three of them give \langle E \rangle = \dfrac{3}{2}kT.
What matters is that no mass appears in it. A heavy molecule and a light one at the same temperature carry the same kinetic energy. The light one simply moves faster to make the account balance.
The figure has the mean energy against temperature, a straight line through the origin. The relation is close enough to say that temperature simply is energy, with k once again doing nothing but matching units. How generally this form holds is the subject of the next piece.
Last comes pressure. Every time a molecule strikes the wall and bounces back, the wall takes up 2mv_x of momentum. How much of that piles up per unit time is the force.
The arrow in the figure is one such strike. The number of molecules reaching the wall per unit time goes as the density times v_x. So the force goes as the mean of v_x^2, and into that we put the previous scene's \langle v_x^2 \rangle = \dfrac{kT}{m}.
Out comes pV = NkT. A gas law found by experiment has appeared from nothing but counting how often molecules strike a wall. It is also a check that everything built so far really does reach the world.