This is the scene from the movie:

[There should be a GIF or video here. Update the app now to see it.]

The first time Rey is shown having the Force was on Takodana, when the lightsaber called her in Maz's palace. It was also the first time we see Rey and Kylo Ren together.
When Rey touches the lightsaber, she saw the memory that had haunted her most of her life, the day that her parents abandoned her.

The Force was showing the pain that she was holding in her heart.
But she also saw the most painful day of Ben's life, the pain that he was holding in his heart. The day that he turned to the Dark Side.

After watching the Last Jedi, we see that this was the night when Luke thought that there was no hope for Ben, so he tried to kill him in his sleep. Then he realized his mistake but it was to late.

The two visions she had of the past was the worst day of her life and the worst day of Ben's life. The days when their family who should have taken care of them no matter what, gave up on them.

(I love this picture for that very reason, it shows both of them in the loneliest moment of their lives. The moment that broke them on the inside. But it also shows them comforting each other in the worst moment of their lives. Thank you to the artist for this beautiful picture!)
In a deleted scene she saw was a young boy, Ben, with Snoke. That was another one of the worst parts of Ben's life.
A Star Wars fact that I read was that Snoke has tormented Ben his whole life, and I think that the Force was trying to show Rey that, in this deleted scene.

The rest of what she saw was more confusing, was it the past or the future?
She saw Kylo Ren and his knights on a dark planet in the rain.

This scene gives us our first glimpse of the knights of Ren.
But there are also a lot of things in this scene that puzzles me.
Was Rey really there, in the future?
Kylo Ren stabs someone, then he looks directly at Rey. But he doesn't threaten her or attack her.
I have seen several posts where people think that Kylo Ren actually saved Rey in this scene. And it really does look that way.

Rather this scene will happen exactly the way it did in Episode IX, I can't say. But since J.J. Abrams, who directed the Force Awakens is also directing Episode IX, he might decide to include this scene. I think it would be cool if he did.
But the future is always moving, just because she had this vision doesn't mean that it will happen. But I do think that it means something. Part of what it means, to me beyond a possible future, was that he would save her, and he did.

Rey also saw Kylo Ren coming towards her in a snowy forest. She was terrified of him and tried to run away which was why her vision ended.
But this was actually my favorite part of this scene, not in the movie but in the novelization.

"I'll come back sweetheart. I promise." Said a voice.
This scene comes right after the scene where Rey's parents abandoned her so, some people confuse which scene this dialogue belongs to.
Some people think that it was her parents, but if you read the whole page, you will realize that this piece of dialogue was in a completely different scene from the one where her parents leave her.
And not only that but if you really pay attention you can tell who the voice belongs to.
"... From in front of her, not far away, came the sounds of battle: the cries of the wounded, the clashing of weapons. Then from behind her she heard another voice.
That voice.
"Stay here. I'll come back for you."
She whirled, glazed eyes desperately scanning the dark gaps between the slender trees, trying to penetrate the darkness."
(So she is now faced in the direction that the voice had come from)
""Where are you?" She started running towards the voice.
"I'll come back, sweetheart. I promise."
"I'm right here! Right here! Where are you?"
No response. She started forward again, running, only to be brought to a sudden halt by a figure appearing without warning from behind a tree."
(The figure was Kylo Ren/ Ben Solo)
"She screamed and screamed again, and fell backward, backward, sitting down hard in-
She was in the underground corridor, sitting on the cold old stone..."

"I'll come back sweetheart. I promise."
Some people thought that the voice was spoken when her parents abandoned her and that it was one of her parents telling her that they would come back for her.
But there is more than one reason that I don't believe that.
First, if you read it, you can tell that this line was spoken while she was in the forest, not when her parents were leaving.
"She whirled, glazed eyes scanning the dark gaps between the slender trees..."

This is the scene from the junior novelization:

She heard someone say "Rey," behind her, she turned to see a strange boy behind her, (which could mean that it was the boy who said her name) I know from a deleted scene that the boy was Ben, when he was young.

These words were in both novelizations, so they must have some significance.
And in both novelizations, these words were spoken in her vision while she was in the forest, not when her parents left. So if this dialogue was not spoken when her parents abandoned her then who spoke them?
The answer is the next scene.
After Rey sees the vision she tries to leave the room as quickly as she can, but she runs into Maz.
And this is what Maz said to her: "dear child, I see your eyes, you already know the truth. Whoever you're waiting for on Jakku, they're never coming back."
"I'll come back sweetheart. I promise."
Whoever spoke those words promised her that they would come back, but whoever she was waiting for on Jakku, her parents, would never come back.
Because Maz spoke these words right after Rey heard the voice say this, it was almost as if she was trying to tell her who the voice belonged to.
"...but there's someone who still could" Maz continued. "The belonging you seek is not behind you, it is ahead."
"Luke?" Rey asked.
Maz didn't answer, but a soft voice did.
"It's Ben."
If you listen very carefully in the video you can hear a voice whisper "it's Ben." Right after Maz says "take the saber."
Rey looks at her like she wants to take the lightsaber, but then gasped suddenly and stood up, she looked down the hall like she heard something. All I could hear on my tv was whispering, but watching it online, I was able to turn up the volume and go back as many times as I wanted. I could actually hear the voice whisper "it's Ben."
The force was trying to tell her that Ben is the belonging she seeks.
And to me this scene of the last Jedi confirms it.

"You're not alone." - Ben Solo, The Last Jedi
Luke never offered her any kind of belonging. But Ben did.
And to me the Force was also telling her that he was the one who spoke these words: "I'll come back sweetheart. I promise."
How does this confirm that he was the one who spoke those words to her?
Well, it doesn't, but this does:
"... she started running towards the voice.
... only to be brought to a sudden halt by a figure appearing without warning from behind a tree."
She ran towards the voice, which means that she was running towards the speaker of the voice. But then after hearing the voice again a figure appears in front of her. So the figure must have been the one who was speaking. And the figure was Ben.
The junior novelization goes into more detail.
"A man in a metal mask, cloaked in black, strode out in front of her..."
Also notice how it is written in the adult novelization: "That voice." She must have heard the voice before, and the voice offered her something no one else had, belonging. "I'll come back sweetheart. I promise." She had been looking for that belonging every since the voice offered it to her, but I don't think that she knew who was speaking to her.
She had been looking for it in her past, she thought that it was behind her but it was ahead.
In the junior novelization: "as in her dreams, she heard no reply."
So she had heard this voice speak in her dreams before.
One of the novels actually mentions one of those dreams:

In the dream there is no figure with the voice, but there is a small clue.
"She'd been haunted by a dream. Or a nightmare."
There are two very interesting points in that line.
One is that she felt haunted by the voice, even though the words were kind and comforting. To be haunted means that she was afraid.
Second is that she said it was a dream or a nightmare: "Turning to run in another direction, she caught herself just in time as a shuttle touched down near by. Without the slightest hesitation the cloaked figure of Kylo Ren emerged and strode forward to join the battle. A stunned Rey could only track him with her eyes. She had seen this man before, in a daydream. In a nightmare." 

It could be a coincidence, but I don't really imagine that most people would look at Kylo Ren and think that they remembered him from a daydream or a nightmare, they would think that they remembered him from a nightmare because he looks frightening. But she remembered him from a daydream or a nightmare, which means that she both feared him and had compassion for him. She had seen both Kylo Ren, in her nightmares, and Ben Solo in her dreams. It is odd that she would describe her memory of him, in the same way she describes the voice, unless he was connected to the voice, or as I personally believe, the voice was him speaking to her.
But secondly, notice this part: "the voice would return, never leaving her alone permanently. Never." 

Not only was the voice in both novelizations, saying the same thing each time, but it even appeared in one of the picture books:

What's interesting about it is that it says "her mind was overwhelmed with visions of Kylo Ren."
That's the only one of her visions that were mentioned, that and the voice that said "I'll come back sweetheart. I promise."
It didn't say anything about her parents. Just Kylo Ren.
When I watched this clip, I listened very closely the whole time but never heard a voice say that line.
But what I did hear was a voice say: "Rey."
And the voice appears in the exact moment Kylo Ren steps out in front of Rey.

Another piece of evidence that supports this theory is that the voice is obviously related to the Force, meaning in my opinion that the speaker had the Force.
It couldn't have been Luke, because he was disconnected from the Force.
I'm sure someone is reading this and thinking that it could have been Leia, but there is a flaw with that as well. The writers say that it was a man's voice.
Luke is the Last Jedi, so that leaves us with the Dark Side. Snoke and Kylo Ren.
I don't think that Snoke would say "I'll come back sweetheart. I promise."
Ben however could have picked up the term sweetheart from his father who called Leia that all the time.

(Thank you to the artist for this great picture!)

All of the evidence that I could find points to Ben being the one who spoke this line: "I'll come back sweetheart. I promise."
Every time she heard the voice in the novelizations, she turned to look in the direction that the voice had came from, she sees Ben, and no one else. To me this means that he has to be the one speaking.