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Verstappen has taken Kvyat's Red Bull seat for the remainder of the Formula One season after the Russian's error-strewn start to the campaign came to a head at his home grand prix when he shunted the team's former star Sebastian Vettel out of the race in Sochi.
Both drivers were present in Thursday's news conference ahead of the Spanish grand Prix, with Kvyat calmly dealing with a barrage of questions on his mental state and future in the sport.
However, the Russian was quick to point out that he had little chance to argue his case for remaining with the senior Red Bull squad.
"There was no real explanation," Kvyat said. "If the bosses want something to happen, they make it happen, simple as that.
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"I think the decision in a way was a bit of shock for myself. It is what it is, I always give my answers on the track and nothing will change.
"I will give as loud an answer as possible on the track.
"I raced for Toro Rosso in 2014, it's a team I really like a lot, and I felt these last few days a very warm welcome, I can already feel the atmosphere is very positive. The goals are clear for the team and myself. I will be pushing the absolute limits on track and will give my answers there."
He added: "I don't see this as blocking [my career], you have to take the positives because there are more positives than you can imagine I think it's a golden opportunity. I believe we can do a really good job, then there is more opportunities to come, you always have something to fight for.
"I think to say dropped is a bit heavy because I still get a good chance from Red Bull with Toro Rosso.
"I think it's a positive thing, I feel like I've done everything for the team. I've been bringing in the points, bringing the development work so I think it's a question for other people who could give a better answer, I don't see any reason, it's a question for them."
Kvyat raised a laugh in the room when explaining the circumstances around his discovery of Red Bull's decision.
"I was in Moscow watching a TV series and then the phone call comes," he explained.
"[I said] 'hello?' 'We have some news...'
"I think it was a 20-minute talk, I said I wanted an explanation, I won't go into details, I'll keep it for myself. Dr [Helmut] Marko made the phonecall and we finished the talk and I went back to bed to finish the TV series, it was Game of Thrones."
Verstappen remained diplomatic throughout, regularly explaining his delight at being able to move his burgeoning career to the next level.
"I'm very happy with the chance they've given me, racing for a top team now that was always the plan," the teenage Dutchman said.
"With the risk, I think it was a bigger risk to be young in Formula One but I've handled it well, now it's just about getting used to the new car which is not easy. The things I've done in the factory have given me confidence it will come race by race and I'm definitely going to enjoy it.
"Daniel [Ricciardo] is a great guy, very fast on track and he has lot of experience in the team, I will try to learn a lot from him."