Lennard-Jones potential and vibrational energy level diagram explanation
This diagram is conceptually horrible, because it is a mix of quantum and pseudo-classical physics.

Forget for a while the vibrations. If you calculate single point energies (within Born-Oppenheimer approximation) for different internuclear values, you'll get something like blue curve.

Now, for the $r_e$ distance between nucleus (again, within Born-Oppenheimer approximation) you can calculate the "vibrational (normally harmonic) frequency" in a pseudo-classic fashion. For this frequency, you can assign energies (of a quantum harmonic oscillator having this frequency). This energy values correspond to the difference between energy values for the horizontal blue lines, and the well (blue).

The widths of the horizontal lines have not physical meaning.

For the sake of completeness, the $D_e$ would be the energy of equilibrium state if no vibrations are considered. $D_0$ when the fundamental vibrational state energy is add. The former energy is not measurable because this is not a real energy for the system.