Your picture is a little incomplete that's all.
But I do not understand why the driving force exists (i.e. why it is favorable thermodynamically for a system to minimize its free energy).

It is statistical  it is easier to go at random to a lower energy than a higher one for a similar reason that it is easier to randomly get a pile of sand than a sandcastle. The "driving force" is kindof a blanket term to cover "whatever it is about the system that tends to shift it from stability" which sounds like a circular definition doesn't it? See below.
I also do not understand why a stable system corresponds to a lowenergy one.

The instability of a system is it's propensity to move away from it's previous state if it is given a little nudge.
A high energy state for a stationary ball would be on top of a hill. Give it a nudge and it rolls to the bottom of the hill  keeps going until it finds a depression, then it kinda rocks back and forth for a bit. So the depression is a lower energy state and stable.
The driving force in this system would be gravity  mainly. The ball won't spontaneously roll to the top of a hill because there are other, dissipative, forces too  it makes a noise, heats up, shifts bits of the scenery it bumps into on it's way, that sort of thing. Taken together, all these things give the ball a "tendency to try to get to the lowest energy state available".
The ball can sit in an identical geometry bowl and be stable at any height though... it may require more than a nudge of energy to get it out of the higher bowl  but once out the ball rolls to the lower bowl (there will be a highly unstable point where it could go either way). The "more than a nudge of energy" would be the reaction energy in chemistry  you have to give paper a bit of energy to start it burning but, once going, it will burn completely all by itself.
That help?
If the best answer would be something along the lines of "take a proper thermodynamics course," that would be fine.

That would help  but even then a decent picture does not emerge until you start to contemplate postgraduate work. You are still at the stage of using the results rather than working the underlying math.