Your question is unclear: energy is defined only to a constant. What matters is how the potential energy changes as the position of the planet changes. So your question might be: why is the derivative of the potential energy zero with respect to the position of the planet?
The planet is like a ball at the very top of a mountain, where the ball cannot decide whether to roll down one side or the other.
That's what the zero of the derivative of the potential energy means. The planet finds both directions (towards star A or B) equally appealing. Zero means it cannot decide between one or the other.
The weird thing is that the planet is onat a saddle pointmaximum (on the line joining the two stars, as pointed out by @Farchersee @Farcher's answer about saddle points). If something happens to nudge one side or the other, then it will suddenly find that it increasingly prefers one star over the other. Suddenly, the derivative of the potential energy is no longuer zero. So, even though the planet cannot decide on its own, where somebody to force the decision for it (say a solar flare from one star), then it's gonna roll with it.
To be accurate, the planet is at the L1 Lagrange point of the system.