How does thermal insulation work at the molecular level? I don't understand why insulation works. For example, imagine we have the same thickness of wood, fiberglass and aerogel. Each insulates to a different degree, the more rarified material, the greater the insulation. I don't understand why this is.
For example, even though there might be more air and light molecules in fiberglass than wood, when a molecule strikes those, the energy is conserved. So, for example, if a molecule of air struck versus a molecule of cellulose, the air might have less mass, but since the energy is conserved, it just means the molecule of air will end moving faster (mv = mv) than the molecule of cellulose. So, I would expect the same amount of energy to be transported in both cases.
What am I missing?
Note that the same exact question was asked on Quora and got only two answers, a hand-waving read-the-textbook answer with no information, and another which said this:

An alternative view; Electric potential (of an atom) is the magnitude
  of angular deflection of its atomic axis, from its natural alignment
  with respect to axes of neighboring atoms, when in electrically
  neutral state. Any property of atom/molecule that reduces ability of
  neighboring atoms to influence relative deflections of their atomic
  axes will increase insulating property of a material. For more
  details, see: Chapter 14 of 'MATTER (Re-examined)'.

I don't really understand this "answer".
 A: 
Each insulates to a different degree, the more rarified material, the greater the insulation. I don't understand why this is.

I will try to explain this part with a very schematic example. 

In the picture above, an air molecule collides with a compact wall, for example a crystal, separating two spatial areas, A and B. Treating the molecules as billiard balls, we will say that the air molecule will hit precisely one molecule of the wall and then bounce away. 
The momentum delivered by the air molecule will result in a vibration of the molecule which was hit. Since the material is dense, this vibration will be immediately transmitted to the nearby molecules.
So, collision after collision, the vibration (temperature!) of the molecules of the wall will increase and increase, until a state of equilibrium is reached. I speak of equilibrium because also the molecules of the wall will transfer momentum to the surrounding air molecules!
So if side A was initially hotter than side B, side B will get hot in no time because of the energy transferred by the vibration of the molecules of the wall.
At equilibrium, for the equipartition theorem, the average kinetic energy of every molecule in the system (side A, side B and wall) will be 
$$\langle \epsilon \rangle = \frac f 2 k_B T$$
where $f$ is the number of degrees of freedom of the molecule ($3$ for a gas particle, $6$ for a particle in a perfect 3D crystal) and $T$ is the equilibrium temperature.
Let's now consider the case depicted below:

The compact wall has now been replaced by two very thin layers containing some gas. You can see that in this case, when an air molecule hits the wall, the vibration is again immediately transmitted to the nearby molecules in the wall, but only rarely to the gas molecules contained in it.
You can then see that this time it will take much longer to get all the gas molecules contained in the wall to receive the kinetic energy and transfer it to the molecules contained in side B.
This is why solid materials have an higher thermal conductivity than gases (and liquids): because when the molecules are far apart from each other it is difficult to transfer energy.
Moreover, ordered materials (crystals) have an higher thermal conductivity than disordered materials (such as glasses), because in the former the vibration can be transferred through well defined directions, i.e. the atoms can vibrate in a synchronous way along preferred directions, while in the latter this is not possible.
Indeed, glass has a terrible thermal conductivity, about 0.8 W/m K. Water and air are even worse, with thermal conductivities of respectively 0.6 and 0.024 W/m K.
Silver, copper and gold are quite amazing thermal conductors, with conductivities of respectively 406, 385 and 314 W/ m K.
But the real hero is diamond, with an astonishing 1000 W/m K: its secret is in its perfect tetrahedral network structure of strong covalent bonds, equals in every direction, which allow it to transfer vibration efficiently from one atom to the other.
