Let the Sun be at the origin. The gravitational force $\vec{F}$ exerted by the Sun on the planet is a central force, meaning it is always directed along the position vector $\vec{r}$. The torque on the planet about the Sun is $\vec{\tau} = \vec{r} \times \vec{F}$. Since $\vec{r}$ and $\vec{F}$ are parallel, their cross product is zero, so $\vec{\tau} = 0$. The rate of change of angular momentum $\vec{L}$ is equal to the net torque, so $d\vec{L}/dt = 0$. This means the angular momentum vector $\vec{L}$ is constant in both magnitude and direction. By definition, $\vec{L} = \vec{r} \times \vec{p}$. The vector $\vec{L}$ is perpendicular to both $\vec{r}$ and the momentum vector $\vec{p}$. Since $\vec{L}$ is a constant vector, the position vector $\vec{r}$ and momentum vector $\vec{p}$ must always lie in the plane that is perpendicular to $\vec{L}$.