In The Last Jedi, both Kylo Ren and Rey see visions of the future that make them think the other one would turn. I could understand that Rey seeing
Kylo kill Snoke might make her think that he will turn to the light.
But what exactly did Kylo see that made him think Rey would
turn to the dark?
It’s that old standby of prophecies, exact words. Both Rey and Kylo are correct in what they saw. It’s important to note that Rey says she saw Kylo’s future, but “just the shape of it” (so she was receiving some impressions, not necessarily images). Rey doesn’t necessarily see Kylo killing Snoke, and Kylo doesn’t necessarily see what Rey will do. Both they and the audience are misled into believing that a certain interpretation of what they have seen is correct.
What Kylo says he saw:
“Because of what I saw, I know when the moment comes you will be the one to turn. You’ll stand with me.”
True. When the moment came (e.g. when the guards attacked) Rey metaphorically stood with Kylo. She also literally stood with him (back to back, I think?) and faced the guards in battle.
Kylo was misled by what he had seen.
He saw Rey standing with him in battle, and assumed that this meant she had turned to his side (or to the dark side). Alternately, like Rey, he may merely have received an impression of the shape of her future, and that impression might have carried the general idea of Rey standing at his side, without being clear about the circumstances.
As a side note, Rey’s impressions are also true:
What Rey says she saw:
“You will not bow before Snoke.”
True-ish. Kylo admittedly does literally bow before Snoke, but doesn’t do so in the sense that matters, since he kills him.
“I’ll help you. I saw it.”
True. Rey helps Kylo in his battle with Snoke’s guards.
But what about Snoke? Is it just that Snoke is deceiving Kylo?
Snoke connected Rey and Kylo, but I don’t believe he gave them the visions. Let’s look at what he says.
SNOKE: Have you seen something? A weakness in my apprentice—is that why you came? Young fool. It was I who bridged your minds. I stoked Ren’s conflicted soul. I knew he was not strong enough to hide it from you, and you were not wise enough to resist the bait.
Snoke doesn’t seem to be talking about Force visions at all, here. He’s saying that he tried to make Kylo more conflicted, and that Rey would be able to see that conflict—and that Rey would notice it and come to him. All of this is true. Note that he doesn’t mention deceiving anyone in any way here, directly. He just let Kylo be himself, and Rey do what she would do. Further, I don’t think Snoke had any role in the contact after initiating it. Indeed, it lingered after he died.
Indeed, if we consider the events of the following film, it appears that Kylo and Rey are a dyad in the Force, and this is related to their connection. As such, it's not clear that Snoke was even being honest about creating the connection, or if he was, it's clear that he exaggerated his role in doing so: at best, he strengthened a pre-existing connection.
In any case, he doesn’t say anything to indicate he knows Rey and Kylo’s visions (which seems like a big omission), and the fact that he knows that Rey came because of Kylo’s conflict is simply because that was his plan. This (along with the fact that the visions all came true, if not in the manner that Rey and Kylo imagined), strongly suggests to me that these were true visions that came from the contact that Snoke created.
In a much later scene, Snoke says something along the lines of “I gave you the visions and connected you two through the force so that you might come here to me.”
Basically Snoke tricked them.
He constructed the visions in their minds so it is nothing more than a mind trick. The visions are fake. Therefore, they don’t have to come true to preserve the continuity of the movie, even if in some way they do.
In that case, it would be Snoke causing his own deciet to come to pass.
There's a larger dynamic at play here. What we see, from our vantage as the audience, is Rey go through
This leads Rey to enthusiastically and willingly go to him, interpreting the vision as Kylo Ren turning back to the Light.
Kylo Ren has no way of knowing any of this. Instead, he's had his own visions of her helping him. Coupling this with the fact that she willingly comes to him, he comes to the opposite conclusion, and that Rey will finally become his apprentice (just like he wanted in The Force Awakens).
I think of two factors:
Add these together, and in general the Sith try to turn Jedi by getting them to tap into the inherent darkness inside of them. They are vulnerable to the pull of the Dark Side by nature, and the Sith can work on them by exploiting their specific, personal weaknesses.
Kylo mistook her joining him when what he saw was Rey and Ben joining forces. As per the The Rise of Skywalker:
You took my hand, no I took Ben’s hand.
