The theory of friction that is described in the source is the Prandtl-Tomlinson model. I'll explain it in two steps to answer your two main questions.
Q1: What is meant by "partial irreversibility of the bonding force?"
All bonding, including the bonding responsible for friction, is due to electrostatic attraction between atoms. Here is what that looks like for Hydrogen:
Image source: Chemistry 301, University of Texas (online resource)
As the two hydrogen atoms approach each other from very far apart, they experience an attractive force until they reach the minimum of the potential energy "well." At this point they experience a repulsive force if they move any closer together. Once the atoms have reached the most stable state it requires additional force to disturb the system, which is the "irreversibility of the bonding force." As two surfaces slide by each other, bonds are being continuously broken and formed in a periodic fashion.
Q2: What does "irreversibility" have to do with friction?
In the Prandtl-Tomlinson model, the force resisting two surfaces moving past each other is modeled as a ball (point mass) climbing up a series of hills due to a horizontal force marked F:
The model helps explain some features of macroscopic friction, such as why there is "static friction" vs. "kinetic friction" -- it takes a little extra force to get the ball moving at first. In the model (details below the jump), increasing the force squeezing the two surfaces together increases the height of the hill. The irreversibility has to do with the fact that it is easier for the ball to roll down the hill than roll up it, and the "degree of irreversibility" is increased if the height of the hill is increased. It is just a model of friction, because friction itself is very complicated. The source you cite connects the irreversibility of bonding and the irreversibility in the Prandtl-Tomlinson model, but that is not stated explicitly in any reference I have access to.
For the curious, here are the details of the Prandtl-Tomlinson model, from an article published by the Technical University of Berlin: