TL;DR: It's specifically because he's a being of logic. Also, because of love.
First and foremost, it's important to remember Vision's original statement of loyalty from Age of Ultron. He is going to side with whichever group will protect life the best.
CAPTAIN AMERICA: Are you? On our side?
VISION: It's not that simple. (pause) I'm on the side of life. Ultron is not.
He explains that he sees logic in the oversight (all following quotes from the Civil War transcript):
Vision: In the 8 years since Mr. Stark announced himself as Iron Man, the number of known enhanced persons has grown exponentially. And during the same period, the number of potentially world-ending events has risen at a commensurate rate.
Steve Rogers: Are you saying it's our fault?
Vision: I'm saying there may be a causality. Our very strength invites challenge. Challenge incites conflict. And conflict... breeds catastrophe. Oversight... Oversight is not an idea that can be dismissed out of hand.
He loves Wanda. Actor Paul Bettany confirms this in an interview with EW. Speaking of why he accidentally blasted Rhodey out of the sky, he says
“What happened there is that his judgment was clouded there by real love and affection,” says Bettany. “And he responded in a quick and thoughtless way.… I really love that the moral compass of the movie is this synthetic person who’s trying to figure out what it means to be human. It’s a neat idea, you know?”
Because he loves Wanda, he believes that this is the best way to protect her.
Tony Stark: If we don't do this now, it's gonna be done to us later. That's the fact. That won't be pretty.
Wanda Maximoff: You're saying they'll come for me.
Vision: We would protect you.
He truly believes that protecting Wanda from the public is also protecting the public.
Wanda Maximoff: Vision, are you not letting me leave?
Vision: [He blocks her way.] It is a question of safety.
Wanda Maximoff: I can protect myself.
Vision: [He holds her arm.] Not yours. Mr. Stark would like to avoid the possibility of another public incident. Until the Accords are on a... more secured foundation.
Wanda Maximoff: And what do you want?
Vision: For people to see you... as I do. [She looks at him gravely.]
Vision: [He faulters.] If you do this... they will never stop being afraid of you.
So, he decides to go the utilitarian route: the greatest good for the greatest number:
Vision: Captain Rogers. I know you believe what you're doing is right. But for the collective good you must surrender now. [Tony's team arrives.]
Of course, being that he is basing his allegiance on logic, he is willing to swap sides.
Actor Paul Bettany addressed this briefly in the interview mentioned earlier.
“He has logic in abundance. And logic does not afford any room for loyalty. New information could come to light and he might flip to Captain America’s side. But love solidifies things and provides loyalty. So if he’s there to protect mankind – or frankly, the universe – he needs to figure out what’s good about these creatures, and I think that’s the quest that you find him on. And, of course, in the comics, the relationship becomes romantic. We’ll see moving forward what happens for Vision and Wanda.”