Noah Schweber said here the following:

Why would you want a notion of sheaf theory for objects more general than topological spaces? Well, the original motivation (to my understanding) was to develop a notion of etale cohomology for schemes; so if you care about schemes, you should care about sites.

Question : In what sense does Grothendieck topologies are in relation with Etale cohomology of Schemes?

Any other explanation for "why would you want a notion of sheaf theory for objects more general than topological spaces?" is also welcome.