What is the molecular interpretation of Raoult's law?
I believe what the quote is saying is that once equilibrium is reached between the vapor-liquid mixture, then there will be no tendency to increase entropy. This is because entropy is not the only factor to consider. In any system, one must always consider the interplay between both entropy and energy. For instance, at equilibrium, $$\Delta G=\Delta H-T\Delta S=0$$ which means that at equilibrium, $$\Delta H=T\Delta S$$So, while you raise a good point that in most cases, a system will tend to maximize entropy, in a closed system, one must consider the interplay between entropy and energy which in this case are exactly equal (technically they differ by a scaling factor which is the absolute temperature).

Additionally, what is meant by "vapour pressure represents the tendency of the system and its surroundings to reach a higher entropy" is that one can make an inference about the energetic and entropic properties of a solution based on the vapor pressure of that solution. For instance, if a system has a very low vapor pressure (consider a solid as an extreme example) then the solution is very energetically stable and thus it has a very low tendency to reach higher entropy. But, if you think about acetone, which has a vapor pressure of $30.6 ~\text{kPa}$ at $25^{\circ}~\text{C}$ (ten times that of water at the same temperature), one could say that this then has a strong tendency to increase its own entropy and the entropy of the surroundings.