TL;DR
Both Plutarch and Cinna had moral sentiments that aligned with those of the rebellion.
Long answer
At the end of Catching Fire, Haymitch explains to Katniss what has happened, and why she had been rescued from the arena.
Plutarch Heavensbee has been, for several years, part of an undercover group aiming to overthrow the Capitol.
But why? Katniss and Plutarch give clues, but aren't necessarily explicit. When Finnick films the lies of the Capitol, for example:
Since my opinion of the Capitol and its noble president are already so low, I can't say Finnick's allegations shock me. They seem to have far more effect on the displaced Capitol rebels like my crew and Fulvia -- even Plutarch occasionally reacts in surprise, maybe wondering how a specific tidbit passed him by.
This insinuates Plutarch had knowledge of a lot of Snow's wrongdoings, and wasn't willing to accept them. Instead, he chose to fight for the government he described to Katniss, Gale and Haymitch with unusual conviction:
"If we win, who would be in charge of the government?" Gale asks.
"Everyone," Plutarch tells him. "We're going to form a republic where the people of each district and the Capitol can elect their own representatives to be their voice in a centralized government. Don't look so suspicious; it's worked before."
"In books," mutters Haymitch.
"In history books," says Plutarch. "And if our ancestors could do it, then we can, too."
And finally, at the end of the series, Plutarch says:
"Now we're in the sweet period where everyone agrees that our recent horrors should never be repeated," he says. "But collective thinking is usually short-lived. We're fickle, stupid beings with poor memories and a great gift for self-destruction. Although who knows? Maybe this will be it, Katniss."
"What?" I ask.
"The time it sticks. Maybe we are witnessing the evolution of the human race."
And as for Cinna, he was never quite normal by the Capitol's standards, especially for a stylist. Most notably he asks for 12, the least desired district, to style in the 74th Hunger Games, but also wears very little makeup (gold eyeliner) and never submits to the same shallow gossip and vanity as the prep team, for example. Everything he does is to genuinely help Katniss (and the rebellion) in the games.
Cinna began the sketches for the mockingjay-soldier costume right after the Quarter Quell announcement, so he had at least been involved in the rebellion that long. Any earlier, he was at least defiant in his own ways if not yet officially involved.
Perhaps one of the best text examples of Cinna's sensibility is during his first meeting with Katniss, when she eats a decadent meal:
I look up and find Cinna's eyes trained on mine. "How despicable we must seem to you," he says.
Furthermore, Cinna unites Katniss and Peeta from the beginning, which is against everything the Games stand for, by having them hold hands during the tributes' parade, which Haymitch refers to as "Just the perfect touch of rebellion."
And of course, turning the wedding dress into a mockingjay is a huge act of rebellion, that ultimately leads to his death:
And suddenly I am so afraid for him. What has he done? Something terribly dangerous. An act of rebellion in itself. And he's done it for me. I remember his words.... "Don't worry. I always channel my emotions into my work. That way I don't hurt anybody but myself.
So Cinna is clearly ready to risk it all for Katniss.
