SÃO PAULO, Brazil — Just a few weeks ago, Mayara Vivian felt pretty good when a few hundred people showed up for a protest she helped organize to deride the government over a proposed bus fare increase. She had been trying to prod Brazilians into the streets since 2005, when she was only 15, and by now she thought she knew what to expect. But when tens of thousands of protesters thronged the streets this week, rattling cities across the country in a reckoning this nation had not experienced in decades, she was dumbfounded, at a loss to explain how it could have happened. “One hundred thousand people, we never would have thought it,” said Ms. Vivian, one of the founders of the Free Fare Movement, which helped start the protests engulfing Brazil. “It’s like the taking of the Bastille.”