Let the object of mass $m$ move with constant velocity $\vec{v}$. For uniform linear motion, the net force $\vec{F}_{net}$ on the object is zero. The torque about any fixed point is $\vec{\tau} = \vec{r} \times \vec{F}_{net}$. Since $\vec{F}_{net}=0$, the torque $\vec{\tau}$ is also zero. Because torque is the rate of change of angular momentum ($\vec{\tau} = d\vec{L}/dt$), a zero torque implies that the angular momentum $\vec{L}$ is constant (conserved).

Alternatively, the magnitude of angular momentum is $L = mvr\sin\theta$, where $\theta$ is the angle between the position vector $\vec{r}$ and the velocity vector $\vec{v}$. The term $d = r\sin\theta$ is the constant perpendicular distance from the point to the line of motion. Thus, $L=mvd$. Since $m$, $v$, and $d$ are all constant, $L$ is constant.