If the pendulums were free & either one were displaced a small distance $x$, the restoring force would be $m{\omega_0}^2 x$. But in the present situation the coupling spring is stretched a distance $2x$ & exerts a restoring force of $2kx$, $k$ being the spring constant. Thus the equation of motion is:
$$ m\dfrac{{d^2}x}{d{t^2}} + m{\omega_0}^2 x + 2kx = 0.$$
Now, the pendulums are coupled. Then why does the equation contain the term $m{\omega_0}^2x$ which is meant for uncoupled condition i.e. when each pendulum is free?