Quick summary: Wakaba Hiiro is actually an alias used by Administrator D, so she could pretend to be a Japanese student out of boredom. The person the story is following, is actually the 'Class Pet': a spider.
The previous Hero and Demon Lord tracked her to this world, but their incompatibility in this universe caused an explosion that killed everyone in the classroom. D was fine, taking responsibility for her actions (and for the fun of it) by reincarnating all the students and their homeroom teacher into this world.
However since Wakaba Hiiro is on the Class Registry, her absence could expose what D had been doing without the other Administrators knowledge. Not wanting to be saddled with more work or talking to, she also reincarnated the spider's soul, hence why she came to this world as a spider. But D implanted her memories and false ones into the spider, as well as the tag of 'Wakaba Hiiro.' This spider (now known as Kumoko) would not only think she's Wakaba Hiiro, but so would the Class Registry and her evolutions with more human appearances. She expected Kumoko to die quickly, but she defied her expectations by cheesing everything and everyone around her, to the point that she was able to break through the System, becomes a Goddess, and now in her new identity as Shiraori has reached where D is.
And why does D let an irregularity like Shiraori to exist? Because she too much fun!