Things aren't always in states of minimum energy. This is something that applies to equilibrium states. Simple examples of this idea are certainly familiar to you already - a ball comes to rest at the bottom of a valley, not partway down the side.
If we want to find the equilibrium state for a white dwarf, there must be no forces on it. If there are no forces, then small changes to the white dwarf don't change its energy, since the change in energy is force*distance. That means the energy should be a "stationary point", where the energy curve is flat. (In this case, we're looking at a curve that describes energy as a function of radius.) We want a minimum so that the equilibrium will be stable.
You can apply different reasoning to get to the same result. For example, the white dwarf is in a cold universe, so thermodynamics says it will give away energy as much as it can since this increases the total entropy. This only stops when the white dwarf is at a minimum energy, or at least gets down to a few degrees K.