What causes contact resistance?
Another term is thermal resistance,
This is incorrect. Thermal resistance is something that prevents heat flow. It is an entirely separate concept from electrical resistance.
How is contact resistance explained?
To obtain very low resistance in a material like most metals, the electrons must be delocalized from the individual atoms, and free flow around in material. When two pieces of metal are apparently touching, they might not be in such intimate contact that electrons can flow freely between them. In fact, if they were we would likely consider them as welded together.
Two touching surfaces might not be ideal. There might be oxides on the surfaces, or dirt.
The two surfaces are not perfectly smooth, so the area of intimate contact is much smaller than the macroscopic area of the two surfaces. Even a a few microns of axial length can introduce measurable resistance if the cross-sectional area is small enough.
What is an intuitive explanation of the sudden loss because of contact resistance?
It doesn't really matter that the interface area is very thin (in the direction the current is flowing) Any situation where electrons must give up energy to pass from one location to another, no matter how thin a region its localized to, will look like a resistor when analyzed as a circuit element.
This is a very interesting question, especially considering the very recent history of scholarship on electrical contact resistance (a term first coined in 1964 by William Shockley, one of the inventors of the transistor), as well as thermal contact resistance. For the following explanation, I will use this research paper on electrical contact resistance published in 1993. In this paper mathematical models of contact resistance for electrical and thermal contact resistance are provided, but here some intuitive explanations are provided.
Now, when electrical current traverses from one medium to another, surface contaminants interfere with the flow of electrical currents. The electrical current must give up some energy to traverse from one medium to another. This can be seen from the model of contact resistance used within the paper.
$$ R_\text{contact} = \left\{ (\rho_1 + \rho_2)(1/[4na]+\alpha^{-1}) \right\} + \rho_f s / A_c $$
Here, the $\rho _f$ is the resistance of the film between the surfaces; $\alpha$ the thickness of the contaminant. The first part of the equation is due to constriction effects, the second due to surface contaminants. Hence, there are two physical effects at play,
Constriction effects enforced when going from one medium to another, which inevitably leads to loss of energy
Surface contaminants which interfere with the flow of electrical current
Indeed the contact resistance can be, as discussed in previous answers, attributed to surface features in terms of asperities and passivating layers. The behaviour of these barriers to conduction depends on contact pressure. Passivating layers are oxides and hydroxides that ubiquitously form on conductor surfaces, and pose a barrier to electron transport. While the presence of roughness features restrict conductance to a limited region of true contact area, the extent of which is substantially smaller than the nominal contact area. Conduction mechanisms through passivated layers (tunneling) nanocontacts at fine asperity to asperity contact (ballistic transport) and the conventional ohmic contacts of larger patches of true contact combine to give rise to the observed electrical contact resistance ECR.
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