Qualitatively, the Morse potential has two competing effects. The first is at small separations, where the potential becomes (infinitely) large; this effect is roughly due to the electrostatic repulsion between the two atoms, and it increases as the atoms get closer together. On the other hand, two atoms may covalently bond, and generally speaking, the energy of electron sharing decreases (is more stable) as the atoms get closer together. To sum up: shared electrons tend to drive atoms together, but electrostatic repulsion at short distances drives them apart. This is the qualitative picture of Morse potential.
The Morse potential is an improvement over the usual first-order approximation in physics, the simple harmonic oscillator (SHO). The wiki page you linked to illustrates the SHO potential alongside the Morse potential. While both potentials successfully model the repulsive response at short distances, the SHO model predicts an increasingly strong attraction as the molecules are pulled apart. Experimentally, this isn't true, and the Morse potential is an approximation that allows the bond strength to decrease with increasing atomic separation.
Other approximate potentials have been used to model the interactions of atoms. The Lennard-Jones (12-6) potential is a simple and popular example. This also has the property that atoms are prohibited from being very close by the rapidly rising potential at small separations, and that bonded atoms are discouraged but not prohibited from being very far away, by a rising but finite potential at long distances. Of course, all of these potentials represent the "optimal" bond length by the distance at which the potential is minimal.
As the temperature goes to zero, the vibrational energy of the molecule tends toward zero. This does not mean that the separation distance goes to zero as well. Rather, the separation distance tends toward the minimum of the potential, at $r=r_e$. It is at higher temperatures that the molecule has enough vibrational energy that it can access shorter and longer bond lengths. At higher temperature, the molecule will have significant kinetic energy if at its equilibrium bond length $r_e$, or significant potential energy if the bond is far from equilibrium. The molecule vibrates, oscillating between large and small separation distances, and passing through the equilibrium separation with maximal kinetic energy. At the longest and shortest separation distances, the kinetic energy approaches zero.
These bond-breaking potentials can predict the highest temperature at which a bond will be stable. The temperature of a molecule roughly dictates how high up the potential well the molecule can climb. At high enough temperatures, the molecule can continue indefinitely toward the right (longer bond lengths) corresponding to the bond being broken. It can also tend toward shorter bond lengths, but since the potential continues to increase at shorter bond lengths, the molecule eventually rebounds, tending back toward long bond lengths and, ultimately, bond breakage.