How do particles lead to momentum diffusion? Because the particles interact, both by sticking together and bouncing off of each other. The tendency of particles to interact can be characterized by a "mean free path" λ, which is the distance that an individual particle travels before interactions totally randomize its direction. There is also an "average atomic velocity" u. With a bit of maths you can work out the relation:
η ~ λ u.
The basic idea is that particles in a fluid only know about other particles within a distance of λ or so, and those particles come in to interact at a rate proportional to u.
A more detailed derivation takes Newton's intuitive idea of two plates being used to shear a fluid, and some force F appears related to the speed v you're shearing them and the distance d they're separated, by the relation F = μ A v / d. We will use this to discover μ.
Now imagine the surface at the top, which is feeling the force. It has some area A and it feels direct collisions only from the nearest λ of the fluid -- the mean free path acts, if you like, as the size of the physically-relevant boundary layer for the plate. All other particles have to bounce around before they hit the plate; only the ones in the immediate λ of the plate can hit it without bouncing off something else first.
As Newton envisioned, the space between the two plates should have a linear velocity profile, so that the average velocity in the first λ is not v, but is instead v [d − λ/2] / d, because we can just peek at the velocity at a distance of λ/2 to get the average velocity between 0 and λ on a linear velocity profile. The difference from the actual velocity of the plate is therefore -½ v λ/d.
So, a bunch of particles come in, but they aren't going at the right speed. The plate needs to speed them up by, on average, ½ v λ/d. That slows the plate down, so the plate needs a constant force to add momentum to keep it moving. That's the core physics idea. The mass of these particles is the only thing missing, but that requires knowing how many particles there are.