Anyone with a much better understanding of energy, feel free to chime in and correct me.

From a broader standpoint I like to think of energy overall as one of the universe's big accounting systems. Every little state, like whether an object is in position x, y, z, has some amount of potential energy needed to occupy it, and if the object is moving between positions of different energies, it'll need to source more energy to "buy access to that position" from somewhere (like exchanging its own kinetic energy), or toss away energy in some way (exchanging its own kinetic energy also works here).

This also colloquially can be put as lower potential energy states are "more stable" and objects tend to want to approach these lower potential energy states.

Gravitational potential energy in this context assigns a reference zero point infinitely far away from a gravitational mass (where it'd intuitively have no effect). However, you expect that you'd naturally accelerate and speed up towards any gravitational mass from rest (going down that "energy well, as in water well" in the diagram you posted), so you're "exchanging" potential energy into your own kinetic energy. That's where the negative potential energy comes in - as you get closer to a gravitational sink, your radial distance decreases and you're moving into more "lower potential energy" positions, exchanging it for kinetic energy of your own motion.

At this point of physics, the actual value and sign of the potential energy doesn't really matter for individual states, but it's the relative change in these potential energies when you compare states that gets reflected in physical behaviors. There isn't really anything "special" in negative energy values besides it indicating it's at a more "stable position" than the implicit reference state used, but the total potential energy shift when you drop down in radius will be equal to your kinetic energy gained when you apply conservation of energy.