According to Han, Supreme Leader Snoke is responsible for Kylo Ren abandoning Luke's tutelage and moving to the Dark Side.
How did he accomplish this?
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According to fully Disney Canon official novelization by Alan Dean Foster, we know 2 things:
Han and Leia suspected that Ben always had a pull towards a dark side, like Anakin. That was a reason he was sent to study under Luke
However, a guess by @user3574984 in another answer is wrong - he didn't just fall to the Dark Side on his own, and then went off and hooked up with Snoke. Snoke had integral part in seducing him to the Dark Side - although at this point, we have no canon information of any level pointing at details of "how".
What could he say to that? What possible response could he give to a willful denial of reason? “He’s gone, Leia. He was always drawn to the dark side. There was nothing we could’ve done to stop it, no matter how hard we tried.” His final words were the hardest to get out. “There was too much Vader in him.”
“That’s why I wanted him to train with Luke,” Leia said. “I just never should have sent him away. That’s when I lost him. When I lost you both.”
Han dipped his head. “We both had to deal with it in our own way.” He shrugged. “I went back to the only thing I was ever good at.”
“We both did,” Leia admitted.
He met her eyes steadily. “We’ve lost our son, forever.”
Leia bit her lower lip, refusing to concede. “No. It was Snoke.”
Han drew back slightly. “Snoke?”
She nodded. “He knew our child would be strong with the Force. That he was born with equal potential for good or evil.”
“You knew this from the beginning? Why didn’t you tell me?”
She sighed. “Many reasons. I was hoping that I was wrong, that it wasn’t true. I hoped I could sway him, turn him away from the dark side, without having to involve you.” A small smile appeared. “You had—you have—wonderful qualities, Han, but patience and understanding were never among them. I was afraid that your reactions would only drive him farther to the dark side. I thought I could shield him from Snoke’s influence and you from what was happening.” Her voice dropped. “It’s clear now that I was wrong. Whether your involvement would have made a difference, we’ll never know.”
He had trouble believing what he was hearing. “So Snoke was watching our son.”
“Always,” she told him. “From the shadows, in the beginning, even before I realized what was happening, he was manipulating everything, pulling our son toward the dark side.”
Note that since it is not directly said in the movies, we can just guess and speculate. I am pretty sure that if there will be a sequel, they will talk his history and how he was caught from the dark side.
What we see / hear in the movie:
The movie doesn't give more tips about his past. Here there is my guess:
It may be an unpopular opinion, but I believe that Han never really understood Kylo and the reason why he betrayed Luke.