Why does the Biot-Savart law give the induced velocity of a infinitesimal segment of a vortex filament at point P?
Why would


  the induced velocity at a point P of an infinitesimal section of the filament be zero unless $d\vec{l}$ and $\vec{r}$ are perpendicular


?

The cross product in
$$
 \frac{ d\vec{l} \times  \vec{r} }{ \lvert \vec{r} \lvert ^3}
$$

does not change signs when going over the $\frac{\pi}{2}$ angle in between vectors, so the infinitesimal filament sections $d\vec{l^-}$ (immediately before $d\vec{l}$) and $d\vec{l^+}$ (immediately after $d\vec{l}$) have contributions to $d\vec{V}$ that do not cancel each other out. Therefore, $\vec{V}$ is not just a function of a single filament section, but of the entire filament, hence the integral.

If you check the next section in the source of your image, titled Straight-vortex case you will see this in action.