Grothendieck's Tohoku paper was an attempt to set the foundations of algebraic topology on a uniform basis, essentially to describe a setting where one can do homological algebra in a way that makes sense. He did this by using the concept of abelian categories. Perhaps a better question to ask yourself is "Why are abelian categories a good idea?" In answering your question, I will do some major handwaving and sacrifice some rigor for the sake of clarity and brevity, but will try to place the Tohoku paper in context.
At the time, the state-of-the-art in homological algebra was relatively primitive. Cartan and Eilenberg had only defined functors over modules. There were some clear parallels with sheaf cohomology that could not be mere coincidence, and there was a lot of evidence that their techniques worked in more general cases. However, in order to generalize the methodology from modules, we needed the category in question to have some notion of an exact sequence. This is a lot trickier than it might seem. There were many solid attempts to do so, and the Tohoku paper was a giant step forward in the right direction.
In a nutshell, Grothendieck was motivated by the idea that $Sh(X)$, the category of sheaves of abelian groups on a topological space $X$ was an abelian category with enough injectives, so sheaf cohomology could be defined as a right-derived functor of the functor of global sections. Running with this concept, he set up his famous axioms for what an abelian category might satisfy.
Using the framework given by these axioms, Grothendieck was able to generalize Cartan and Eilenberg's techniques on derived functors, introducing ideas like $\delta$-functors and $T$-acyclic objects in the process. He also introduces an important computational tool, what is now often called the Grothendieck spectral sequence. This turns out to generalize many of the then-known spectral sequences, providing indisputable evidence that abelian categories are the "right" setting in which one can do homological algebra.
However, even with this powerful new context, there were many components missing. For instance, one couldn't even chase diagrams in general abelian categories using the techniques from Tohoku in and of itself, because it did not establish that the objects that you wanted to chase even existed. It wasn't until we had results like the Freyd-Mitchell embedding theorem that useful techniques like diagram chasing in abelian categories became well-defined. Henceforth, one had a relatively mature theory of homological algebra in the context of abelian categories, successfully generalizing the previous methods in homological algebra. In other words, we have "re-interpreted the basics of [algebraic] topology" by allowing ourselves to work with the more general concept of abelian categories.
Best Answer
As requested:
By Theorem 1.10.1 in Tohoku, an Grothendieck abelian category has enough injectives. Sheaves of abelian groups on a Grothendieck topos form a Grothendieck abelian category. By Theorem 2.2.2 in Tohoku, one may then take derived functors of global sections.
As mentioned in the comments, though, sheaf cohomology on toposes is developed quite a bit elsewhere (e.g. in SGA 4).