According to Aschbacher's theorem, if $H$ is a maximal subgroup of $SL(n,q)$, then either it belongs to one of the classes $C_1 - C_8$, or $H$ is absolutely irreducible and $H/(H\cap Z(SL(n,q)))$ is almost simple.
I know that representatives of the classes $C_1 - C_8$ are described for $SL(n,q)$ in the general case, for example, in the book by Kleidman and Liebeck "The subgroup structure of the Finite Classical Groups".
However, are there any results concerning subgroups of the latter type for the group $SL(n, 2)$, and if so, where can I read about it?
I will be really greatful for this information.
You cannot hope for a general classification of maximal subgroups of this type. For that you would essentially need to find all absolutely irreducible representations over ${\mathbb F}_q$ (with $q=2$ in your case) of all nearly simple finite groups.
There is an upper bound on their order, which is often useful, and is used extensively in maximality proofs: either $H=A_m$ or $S_m$ with $m= n+1$ or $n+2$, or $|H| \le q^{3n}$. This is due to Liebeck, and is Thm 5.2.4 of the book by Kleidman and Liebeck.
There is also a complete classification of maximal subgroups of classical groups for dimensions up to $17$. The dimensions up to 12 are listed in the tables in my book with Bray and Roney-Dougal, and the dimensions from $13$ to $17$ are covered in two PhD theses.