This is a profound question, and at the heart of an active field of research: https://en.wikipedia.org/wiki/Tribology.
In fact, the notions of kinetic and static friction aren't always well defined, and even when they can be defined, they do not always 'adhere' to the simple rules that are taught in physics courses.
The model that I describe here really only qualifies as an explanation, or a sort of mythologizing, which should not be mistaken for a true, fully fledged theory. For an explanation to be a theory, it must also come with (and be able to make) a set of predictions, and have what one might call a domain of relevance (i.e. a theory doesn't have to be all-encompassing, but it can't be vacuous either.)
We begin, of course, with jello (or agar, for vegetarians.) Readers familiar with jello (or agar) may appreciate how a block of jello can be bent and twisted in various ways, and still return to its original shape (maybe after a brief period of undulation.)
This is a non-trivial claim, but essentially all solid objects, no matter how brittle they may be, are able to bend to some extent, even if only very slightly, in a way analogous to how jello bends. This 'bendibility' is partly why solid objects made of metal and wood, like bells and marimbas, produce sound when struck, and turns out to be essential in explaining kinetic and static friction.
Readers may also have at some point touched jello with a cold metal spoon, and noticed how the jello sometimes sticks to it (although the effect can be quite subtle.) Alternatively, readers may have observed a balloon, having recently been rubbed against their hair, subsequently attach itself to a wall. This 'stickiness' between surfaces also turns out to be a universal property of any contact (and partly explains how balloons acquired electrons from adjacent hair in the first place.)
The antepenultimate ingredient that we need is that all surfaces have microscopic features that can't be resolved without a microscope. These features may stick out in various directions, and often cause the surface to be rough when viewed under a microscope. Interested readers are invited to check this for themselves (a surprising amount of magnification can be achieved using clear marbles, or if necessary with a digital microscope [which can be found relatively easily for less than $40].)
Lastly, note that often a significant amount of heat is generated when objects rub against each other, like when you rub your hands together to make them warm (or in my case at least when shivering in the cold.) Where there's heat, there's usually some kind of equilibrium being unsettled, and heat and equilibrium are the two final ingredients that will allow us to explain kinetic and static friction.
With the above concepts in mind, let's imagine what happens at the interface between two solid objects that are pressed together. It might help to think of the interface as a sort of channel through which 'force' (viewed as a sort of 'liquid momentum') flows from object to the other. The objects are pressed together, so the surface features of one press against those of the other, and because they can bend, these features tend to do so (sometimes breaking, allowing new contacts to form) until they are able to balance the pressure. If the objects are held together so that static friction applies, the interface has time to approach equilibrium. Because the surface features are so tiny, this takes little time (for comparison, a warm [not hot] surface can often impart heat before your hand can feel it.) The approach to equilibrium in this case involves a process of relaxation, like the gradual compression of leaves on the ground in autumn, or the settling of a marble at the bottom of a bowl, creating new contact points that bridge the interface, as well as the dissipation of excess heat, whether that heat is in the form of atomic vibrations or undulations of one surface or the other, again allowing more contact to be made. In contrast, if the objects are rubbed against each other and kinetic friction applies, the interface does not have time to reach equilibrium, and the motion of rough surface features against other rough features causes undulations and vibrations that may reduce the degree of contact. Because contact is sticky, this would tend to reduce the associated force along the surface (relative to the static case.) Because more contact is needed to accommodate a larger normal force (a larger channel is needed to accommodate a larger flow), it makes sense that the number of microscopic contact bonds would increase with a larger normal force, producing greater kinetic friction and a larger static friction threshold. As for why the rate of increase is linear, it might be worth thinking about the world of difference between nature at human scales and nature at microscopic (or nanoscopic) scales. To start, it might help to consider the magnitude of forces experienced by electrons in, for example, a hydrogen molecule.
As a way to explore this topic in more depth, interested readers are invited to consider and ponder the following questions:
- Under what conditions might the linear empirical laws of friction be inaccurate?
- What if the contact is intrinsically sticky, as with duck tape?
- How might the empirical law of friction be different if the contact is across a loose arrangement of ball bearings (and the ball bearings are ignored?) Note that at a fundamental level, the ordinary rules of friction still apply here; they are merely 'hidden' by clever and somewhat painstaking precision engineering.
- (Challenge) What if the surfaces in question are vibrating (when emitting sound, for example, as in a squeaky wheel?)