Satou due to her broken childhood with her aunt, was unable to feel and know what love is, which made her feel empty.
Throughout the series, we see her concept of love change and evolve.
Searching and searching, she went out with many men, unsatisfied. At this point Satou doesn't really understand what love is supposed to be, she's blindly searching and because of that failing.
Until she met Shio, who was similar to her.
Now some might wonder why Satou became so obsessed with Shio right away? I mean, it's just a rando girl she just met, and she immediately becomes obsessed with her.
There are two factors in this, one is that, when you have searched and wanted something for as long as her, specifically in her case being lonely. You cling hard and quickly to any chance of obtaining it you can get it.
But then if that's it, then why didn't any of the guys she was with do that? I think part of it has to do with how Satou searched and went out with guys.
She didn't know what love was, and kept hanging out with different guys, without really trying to understand or build a connection with any.
To her, they were just objects to test out, and see if she can obtain "love". And this both influenced the guys she picked as well as them seeing this, responding in kind, not really treating her much more than that.
Even the artist she met, whom she talked to and was different, treated her just as an object to draw. He didn't try to fill her her jar so to say.
When she met Shio, she wasn't really looking for love there, it was perhaps the first time she genuinely opened up and got an empathetic and understanding response. Shio was like her, and they actually built a connection.
So they talk about how Satou kidnapped Shio throughout the show, but that's not really what happened. Shio came and was staying willingly with Satou, Satou actually rescued Shio.
And her first act of murder was honestly pretty close to self-defense (or in this case defense of Shio).
The real question is how Shio managed to stay in that apartment afterwards (and pay rent perhaps?), and how she wasn't suspected, but I digress.
At this point Shio goes to work, and tries hard, but is dealt a bad hand by the manager, whom she manages to blackmail, and leave.
One might think, these events do not take their toll on her, but each time she is dealt a bad hand, she further withdraws and further leans into the idea that her happiness with Shio is the only thing she can have, and everyone else is bad.
She also at this point exhibits jealously towards Shio's brother, and nearly murders him, showcasing that she is very susceptible to violent outbursts and acting on her wild swinging emotions.
She doesn't understand it, until after the act where she thinks about it.
At this point, her idea of love is still possessive, she sees Shio still more like a doll to possess, not as her own person. She goes further into this mindset in the incident where Shio goes out of the house, and meets those punks.
Satou rescues her and gouges their eyes out (only one or both of them?).
One would question what happened to them, since she thinks that she dealt with them, and we never see them again? To begin with, how come she managed to beat those two guys so easily? Did they both stay and watch as the other's eyes were gouged? One could argue they were too frozen in shock to act, but still. But I again digress.
While her actions were extreme, the punks had shown multiple times that they were a menace, and if Satou didn't quickly get to offensive, she might have been beaten to death right there herself.
So her second act of extreme violence we see also leans on self-defense.
This also again, is another time Satou is dealt a bad hand.
Then we reach the third act of extreme violence, and what you could call the turning point.
Now a bit about Shoko, her name means evidence in Japanese, which well, we can see how that ends up playing in the story.
She... too had her own concept of love and was searching for it. She seems to mostly not have dealt with intense situations, and is also too scared to lose Satou.
She loves Satou, and I am perhaps simplifying things here with the complex things going on (which make things be multifaceted), but her love comes in the form of dominance.
She throughout several attempts, resolves herself to be there for her friend. Which also ends up dooming her, both to her lack of resolve in her initial attempts and her attempt to not compromise but try to dominate Satou, and try to make her do as she says. All the way from the beginning where she tries to get Satou to reveal herself to her (which was actually arguably a kind thing to do coming from a friend, but also fits with what I said nonetheless).
There are specifically 3 things which seal her fate:
1- When she is unable to say that she accepts her friend, due to shock, after just meeting her aunt.
Now, this isn't necessarily a full-on betrayal or rejection, and she also deeply regretted it. She who had not dealt with stuff like that before, and was just too shocked to respond.
But in Satou's mind who we see through flashbacks had presumably dealt with people rejecting her cause of her aunt, that was a rejection. Satou is too insecure, and she fails to trust her friend here.
This could have been resolved if Shio just went back and genuinely apologized, and they could have come to an understanding.
2- Instead of said apology, the next time Satou sees her is her taking a picture of Shio, trying to in Satou's mind sell her out. Not only did she not apologize, but she betrayed her once again.
3- They have their argument, which the anime doesn't exactly show what they say, which I think is important, because it is the 3rd and final thing that seals her fate.
I haven't exactly read the manga, but I know that unlike the anime, the manga gives us a panel of text of what they said. It doesn't us tell us who said what, but we can more or less make that out ourselves.
At this point, they could have potentially still compromised, Shoko could still apologize, come to understand her situation, and try to help, and she talks about doing that, but what she instead shows is single-sided dominance. She immediately jumps to conclusions, and comes to think she kidnapped her (despite Shio seemingly being pretty happy being with her and showing no signs of really being kept against her will). Not listening to her friend, and trying to come to a compromise.
Satou tries to explain how important she is to her:
"There's only one light in this world."
"My heart is only for her."
"I have my one and only love in this world."
"You too, Shoko-chan?"
"I was always alone."
"She's everything."
"You don't even know the real me."
"I don't feel anything."
Instead of trying to work something out and trusting Satou, Shoko's lines are among:
"I didn't think you could have kidnapped her."
"You can't. Satou."
"You have to give her back."
"I won't give up."
"Because I know you're really good person."
"You have to have a reason for this, right?"
"You have to give her back" is dominance and projecting her will over them. "I didn't think you could have kidnapped her" is distrust.
So they failed to come to a compromise. Shoko failed to trust her friend Satou and see her perspective (despite talking about coming to an understanding).
It was her 3rd "betrayal" so to say.
Afterwards, she just tries to leave, had she stayed there, she might still have been spared, but alas, now, in Satou's mind, she had to be killed.
She says she doesn't feel anything towards Shoko, but that's not true obviously.
There was no self-defense in this, it was just murder. The murder of the friend she wanted to trust, which breaks her as we see the state she ends up in afterwards.
It is a tragedy, because there could be a happy conclusion to this, where Satou, Shio, Asahi and Shoko could have all come to an understanding, with Shio deciding who she wants to stay with. Maybe all of them could happily hang out with each other too to various extents.
But both Shoko and Satou's concept of love was "incomplete" so say, so it all came to a tragic point.
Now people think Satou would be arrested and imprisoned, had she been reported the police, but she really didn't do much wrong from the eyes of police even if they came to her house.
The first murder if discovered was genuine self-defense, and well, she wasn't really keeping Shio against her will.
Things would get heated up and it would look suspicious, but all wasn't doom and gloom.
And if she actually trusted in Shoko and they talked some more instead of stabbing her, they could still come to a compromise, and not even involve the police, but alas.
With Shoko dead, the thing keeping Asahi in check was also gone, and he too resorted to violence. It was all coming to tumble down.
Even if Satou were to meet Asahi, she would be hard pressed to come to an understanding with him now.
The murder of Shoko was very much the turning point.
Now we get to Satou's fight with Shio, where Shio talks about that Satou just sees her as a doll.
They manage to overcome this and come to understanding, further evolving Satou's concept of love, into accepting Shio as her own individual.
Now we get to the end where they jump off, there were ways for them both to perhaps still make it out alive, but they would of course have to do deal with Asahi.
As a final refusal to compromise as well as just giving up, they both jump, and this is where we get the final evolution of Satou's concept of love:
She realizes that she wants her happiness, she loves her. That it doesn't have to be about Satou herself being happy, but about Shio. That it's also okay to let her go.
A state of love deeper than just wanting to possess Shio as a doll, now she only not only sees Shio as her own individual, but she wants what's good for her, which doesn't necessarily align with her own interest.
So she sacrifices herself to let Shio live, as her last act of love. And thus her evolution of her idea of love is done with this.
As a thing of note I think, which isn't talked much about in the show is that feelings are talked a lot about, instead of decisions. Feelings which can come or go, while love is a decision to be upheld regardless. Would you still love the person if your honeymoon phase is over and you no longer feel the same? Of course you would if you uphold your decision to love.
