I am not an expert in such fields, but I'll give you an overview of how I've learnt it.
The main point to realize is that, on a microscopic scale, the surfaces we initially thought of as "smooth" contain actually a great many irregular protuberances.
Coming back to the surface area between the two objects, one must carefully distinguish between the microscopic area of contact and the macroscopic upon which the friction force is independent, meaning they can be lying on top of each other with their larger cross sections or their smaller parts, it will not matter. Of course this seems surprising at first because friction results from adhesion, so one might expect the friction force to be greater when objects slide on their larger sides, because the contact area is larger. However, what determines the amount of adhesion is not the macroscopic contact area, but the microscopic contact area, and the latter is pretty much independent of whether the objects lie on a large face or on a small face.
Key idea is that the normal force puts pressure on the protuberances of one surface against those of the other which causes the protuberances/junctions to undergo a certain flattening (elastic deformation e.g.), and this increases the effective area of contact between the "rough" parts (before, you can imagine that only the tip points where actually bonding), as illustrated in these two pictures:
Second picture: larger effective area of contact or in other words higher number of contact points between the protuberances, also as pointed out by Jim.
To conclude, we now can tell that for large macroscopic contact areas, the number of protuberances in contact is larger but since the normal force is distributed over all of them, their deformations are less important (smaller effective microscopic area), whereas the opposite will hold for smaller macroscopic surfaces, where the deformations are very strong and maximize the contact between the junctions, but their numbers is comparatively lower. All of which explains why macroscopic areas don't matter.
As for larger normal forces, it will increase the deformation of junctions and make the coupling between the surfaces stronger.
In the extreme case of a gas, viscosity comes mostly from the diffusion of molecules between layers of the flow (this diffusion transports momentum between the layers) — this is why a hard sphere gas is viscous for example. In the opposite extreme of the friction between two solids, this clearly does not happen: in that case, friction comes mostly from short distance repulsion between molecules or atoms.
The problem to go further is that there is a continuum of material between solids and gases, not to mention that a thin layer of the surface of a solid can melt under friction.
Best Answer
I will answer your question by first explaining why fluid friction is velocity-dependent. The resistance between fluid is called viscosity. It is a property arising from collisions between neighboring particles in a fluid that is moving at different velocities. When a fluid is forced through a solid, say a tube, the particles which compose the fluid generally move more quickly near the tube's axis and more slowly near its walls. Therefore, there will be a friction between particle layers, and the friction will be larger if the relative velocity is larger.
For friction between solids(called dry friction) however, because solid particles are bound by electromagnetic force in fixed positions, friction only happens at the surface of contact, and is independent of the relative velocity.