Since he's the son of murderers, Tsumugi is ignored and ostracized by the class. However, it is at least partially his fault, as explained later in the story: he acts cold and distant because that's what people expect of him. He's just playing the role of "the son of murderers".
One day, Tsumugi hears a voice from the girl known as Maya. In truth, she's not actually human and no one else can see or hear her at that point. However, when he says her name is Yamada Hanako, he is enforcing his perception of her and literally defining her existence. She has a combination of "the power of words/names" and "doesn't exist until observed" thing going on. After Tsumugi defines her, Maya is able to be observed by others in the same way as her perceives her. That's why she speaks to him and then she transfers into the class the next day; she didn't exist as a human girl before then. (Never mind the details of how she managed to transfer into the class...)
When Maya is trying to find out more about why Tsumugi is treated as he is, she speaks to Erina, who is the only one who will talk about Tsumugi. (Later, it may have been suggested that Erina's machinations are the reason no one talks about Tsumugi, though I may have misinterpreted that.) When Erina later approaches Maya, Maya pisses her off by both not remembering who Erina is and stating that Tsumugi is important to her. The reason this bothers Erina—and how Maya is "obstructing" her—is because Erina uses Tsumugi's existence to feel a little better about herself; she's under a lot of pressure trying to live up to other's expectations, so she the fact that she's not in Tsumugi's position—where he's completely unimportant to everyone—gives her some comfort. Maya both saying that Erina doesn't matter and that Tsumugi is important to someone shakes her foundation, causing Erina to snap.
First Erina approaches Maya and ends up stabbing her in the eye. Then Erina approaches Tsumugi and while she's assaulting him, he ends up realizing that she's been defined by the expectations of others, just the same as him. Maya intervenes, saving Tsumugi from being stabbed. However, when she's about to "criticize"—read: beat up—Erina, Tsumugi stops her and instead talks Erina down. He causes her to realize that she's letting others define who she is and how she should behave, and that the two of them aren't that different in that regard: they're just acting on other's expectations of them. She's shocked to realize that she's so similar to someone she looked down on and thought of as the complete opposite of her, and so runs off. (She also probably upset at the realization about how much she's letting others control her life.)
Tsumugi then confronts Maya, who explains a bit about what she is. She's related to the monster "nue" somehow. (A nue is a monster said to have characteristics of the animals she named, other than the high school girl.) To be honest, I'm not sure if she's supposed to be literally a nue, or if what people identify as a nue is just an amalgamation of previous forms she's had, if she's just been mistaken for a nue, or if there's some other explanation, but the important point is that she's not human. Every other person who has been able to perceive her saw her in a monstrous form and so defined her existence as a monster. Since Tsumugi saw and named her as a human girl, however, she's able to interact normally with others, which pleases her. She'll remain a human girl as long as Tsumugi acknowledges her existence, so she basically asks him to live for her, albeit in a more sinister sounding fashion.
Later, when Tsumugi and Maya are eating lunch, Erina approaches them to apologize. She's realized the truth of what Tsumugi said—that she's letting others dictate her life and personality—and wants to change. She wants Tsumugi to help her change (somehow?) and offers to stand up for him as well, so that the two of them can develop healthier personalities. When Maya comments that Erina is just trying to use Tsumugi, the two start bickering, because love triangle, I guess.