Let $O$ be the intersection between line $A_1A_2$ and the directrix. From the similitude of triangles $OA_1A_1'$ and $OA_2A_2'$ one gets:
By the converse of exterior angle bisector theorem we then obtain that line $FO$ is the bisector of external angle $\angle A_1FB_2$ of triangle $FA_1A_2$.
The same reasoning can be repeated for triangle $FB_1B_2$: the line joining focus $F$ with the intersection of line $B_1B_2$ with the directrix is also the bisector of exterior angle $\angle B_2FA_1$. But those exterior angles coincide and have thus the same bisector: it follows that line $B_1B_2$ also intersects the directrix at point $O$. That completes the proof.
The argument given above can be expanded a bit, to prove some interesting properties of the tangents of a parabola.
If we let $A_2\to A_1$ in figure above, then lines $A_1A_2$ and $B_1B_2$ become the lines tangent to the parabola at $A_1$ and $B_1$ respectively, endpoints of a focal chord (see figure below). The results proven above still hold, hence those tangents meet at a point $O$ on the directrix, and line $FO$ is the bisector of $\angle A_1FB_1$, i.e. $FO\perp A_1B_1$.
It follows that triangles $A_1FO$ and $A_1A_1'O$ are congruent, implying that the line tangent at $A_1$ is the bisector of $\angle A_1'A_1F$, a first well known result:
The line tangent to a parabola at a point $P$ is the bisector of the
angle formed by the line passing through $P$ and the focus, with the line
through $P$ parallel to the axis.
Finally, from $\angle A_1'A_1F+\angle B_1'B_1F=180°$ one gets
$\angle OA_1F+\angle OB_1F=90°$ and consequently $\angle A_1OB_1=90°$,
which is another famous property:
The lines tangent to a parabola at the endpoints of a focal chord
are perpendicular and meet on the directrix.