To recap: the neuron membrane is semi-permeable, and is more permeable to K+ than it is to Cl-, Na+, and Ca2+. Leak channels give rise to the selective permeability of K+, as they allow potassium to leak out of the cell. At rest, K+ is more concentrated in the cell due to its enhanced permeability (amongst other things), which means that the RMP (-65 mV) is closer to the equilibrium potential of K+ (-80 mV) than it is to the equilibrium potential of other ions, say Na+ (~ +60 mV). (Remember, there is still Na+ and Cl- inside the cell, so that would be partially why the RMP is not the same as potassium's equilibrium potential.)
The Na+/K+ pump accounts for <<10% of the concentration gradient (per Charles' lectures from PHYS20008). However, the ATPase helps establish the electrochemical gradient (and therefore potential energy) that ion exchangers/transporters use.
Even if an ion is freely permeable, the majority of the RMP is established by the leak of ions, and the cell’s inherent negative charge holds the remaining cations in the cell (again, from PHYS20008). From memory, the negative charge comes from some of the structural proteins that comprise neurons.
Since K+ has an equilibrium potential of -80 mV and is a positively charged ion, K+ wants to move into an area of that potential. As -80 mV is the potential required to keep K+ at equilibrium, K+ flows out of the cell until an RMP of -65 mV is established. Once at RMP, K+ is in a relative state of equilibrium (even though the cell is not at -80 mV, K+ leakage out of the cell means that there is less K+ concentration in the cell, and therefore is effectively at potassium's equilibrium potential).
In sum: the slide is saying that the RMP of neurons results from the selective permeability of K+, as well as potassium's equilibrium potential, therefore implying that K+ has minimal net movement as it is close to its electrochemical equilibrium. And yes, the electrochemical gradient keeps K+ in the cell at a certain concentration (or at an RMP of -65 mV).
Please let me know if I was unclear.