He saw her helping him, and assumed she was on his side
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:
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.
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.