No surface is perfectly flat, there's always some roughness. If you look at a flat hard surface with a microscope, at some high magnification level it's probably going to look like sandpaper with lots of tiny bumps (asperities).
So an intuitive, math-free model for two hard surfaces in contact would be: imagine two sheets of sandpaper on top of each other, with the sand covered faces in contact.
Static friction: when there is no movement the asperities on both surfaces can interlock.
Dynamic friction: when one surface is sliding on top of the other, asperities will bump into each other but they have less time to find a solid interlocking position.
This explains why dynamic friction coefficient is lower than static friction coefficient.
In both cases, the harder the surfaces are pressed together, the more friction there will be. So the maximum force due to friction will be proportional to the force pressing the two surfaces together (like the weight of the object if it is on the floor).
Then, as a model, we multiply this by a "coefficient of friction" which depends on surface roughness, material, static or dynamic, etc.