The full question:
Let $SL_2(F)$ act naturally on the projective space $\mathbb{P}(F^2)$ over the field $F$ of at least 4 elements. Show that the stabilizer group $Stab_{[1:0]}$ of the projective point $[1:0] \in \mathbb{P}(F^2)$ is a maximal subgroup of $SL_2(F)$.
This was in the chapter of trying to prove the simplicity of $PSL_n(F)$, the projective special linear group defined as $SL_n(F)/Z(SL_n(F))$. I found this question very difficult and would appreciate some help. I also do not quite understand what they mean by "act naturally".
I am in my first year of learning abstract algebra, and haven't learned about Galois theory yet. This is the first time learning about projective spaces, but I think I understand that lines through the origins could be viewed as equivalence classes represented by points.
edit: Also, I would like to ask, what is the point of this question? Does it help us arrive at the conclusion that $PSL_n(F)$ is a simple group?