Angular momentum of a particle rotating with a central force is constant due to:

A central force is a force that is always directed towards or away from a fixed point (the center). Mathematically, $\vec{F}=F(r)\hat{r}$.The torque $\vec{\tau }$ about the center is:
$$\vec{\tau }=\vec{r}\times \vec{F}=\vec{r}\times [F(r)\hat{r}]$$
Since $\vec{r}$ and $\hat{r}$ are in the same direction, their cross product is zero.
$$\vec{\tau }=0$$
By the relation $\vec{\tau }=\frac{d\vec{L}}{dt}$, if torque is zero, the angular momentum $\vec{L}$ is constant.