It seems as if people are turning out in huge numbers just for the primaries
It may seem like that, but it's not entirely true. We have roughly the same turnout this year as we had in 2008. The main difference is that Democrat turnout is down relative to 2008 (and the historical average) while Republican turnout is up. Of course, 2008 was a record setting year. If you compare to other years, overall turnout looks better.
A side issue is that Republican primaries are often decided by this point. Yet this year, only one candidate (Marco Rubio with 71% in Puerto Rico) has managed to win the majority of the vote in any primary. This tends to increase turnout, as these later states have more reason to vote.
In terms of reasons why anti-establishment candidates are doing well now, remember the problems that have persisted since the Clinton administration. While ISIS has replaced Al-Qaeda, jihadist terrorism remains a persistent worry (even though 9/11 was the only successful large scale attack). Many Americans are frustrated that politicians don't seem willing to take this seriously.
There's also the ongoing deficit problem. In Barack Obama's eight years as President, the deficit spiked and is now back where it was at the end of George W. Bush's term. Debt remains stubbornly larger than GDP.
Culturally, the country is changing. Twenty-five years ago, there were still laws against homosexual acts. Now, gay marriage is a court-defended right and a prominent transsexual is being asked who is the best Republican candidate.
A substantial portion of immigration is illegal. People just walk across the border. This doesn't satisfy anyone. The people who come here illegally aren't satisfied, as they have to undergo a risky journey and can never visit home without repeating it. The people who want someone to come here legally are dissatisfied. There are so few slots remaining that there is always pent up demand. The people who don't want immigration aren't satisfied, as there is still immigration.
We've been at war since 2001. But there are few victories or achievements. We killed a terrorist leader and a new organization replaced his. We may have fewer soldiers in Iraq and Afghanistan, but it doesn't feel like we won.
While overall compensation including benefits have increased, it's been decades since wages have. Compensation has instead come in the form of increased benefits, e.g. health care. Wages have remained stagnant.
Unemployment statistics have dropped, but we haven't had the normal post-recession surge of growth that powers higher wages. We're not exactly in recession, just a sort of stagnation. Bad jobs abound while people can't find good jobs to replace them.
For eight years, there's been a black progressive as President, but there's been minimal progressive change. No universal healthcare, just expanded Medicaid and subsidized private insurance. No new taxes on the rich, just an expiration of some of the tax cuts from ten years ago. No big new infrastructure spending, just a couple years of stimulus at the depths of a recession. No break up of the big banks, just a new bailout fund and some new regulations.
For six years, Republicans have controlled the House, but no tax cuts. No regulatory cuts. No spending cuts. No entitlement reform. Even sequestration's reduction in the rate of increase has been ended. Obama was still able to reopen trade with Iran and Cuba. Foreign-supported terrorist attacks continue.
This has been building for a while. This year's explosion of candidates (twenty-two between just the two major parties) has exposed this more than previous elections did. It was probably equally true in 2012, but there was a clearer sense of it being Mitt Romney's turn then. In 2008, the comparatively anti-establishment Barack Obama beat Hillary Clinton and maverick John McCain topped the Republicans. In 2000, Pat Buchanan did about as well as Trump is now, it's just that George W. Bush did better.
Or realize that they aren't doing well. Bernie Sanders continues to underperform his delegate targets. For example, he needed 75 delegates from Michigan but only got 65-72 (final count isn't released, but Hillary Clinton got at least 58 of 130). The big news there is that he didn't underperform by as much as the polls predicted that he would. And Clinton still had the expected blowout in Mississippi, so Sanders is farther away from the nomination than he was the previous day.
Donald Trump has a perceived image as an anti-establishment candidate, but until recently he was part of the establishment. I.e. when they talk about billionaires owning establishment politicians, he was one of the billionaires. As the recent New York Times leak makes clear, in private, he's telling people to ignore what he says publicly about immigration, trade, and torture. And he's not doing that well either. He's actually behind Sanders in percentage of the vote (different parties). The difference is that the more fractured Republican field leaves him as doing better than anyone else while Sanders remains in second place.
Like Sanders, Ted Cruz is a sitting Senator, so a bit of a reach as an anti-establishment candidate. He can point to a certain amount of obstruction that has effectively made Congress act more Democratic than it would have otherwise. Apparently it is anti-establishment to empower the establishment. He remains behind Trump, and continues to lose states he should be able to win (e.g. Mississippi).
Rubio has been embraced by the establishment since his anti-establishment win in 2010. He's not doing well by most measures. Although, as I said previously, he is the only Republican candidate to win the majority of the vote in any primary. He used to have a high favorable rating, but that has dropped recently.
John Kasich is the most establishment of the remaining Republican candidates. He hasn't won a single primary. So if you want to point at establishment candidates doing badly, there he is. Obviously, Jeb Bush and Martin O'Malley already dropped out.
Ben Carson has the best anti-establishment credentials of any candidate, but he dropped out without winning a primary or even making it into the top three.
Hillary Clinton is a clear establishment candidate. She's actually doing the best of any candidate in either party. She's regularly hitting her delegate targets and getting a majority of the vote. Not always, but enough that if she continues to perform at this level, she'll win the nomination.
We perceive some candidates as being anti-establishment. Why? Because that's the way that the media describes them. The leaders in both party nominations are actually quite establishment. In fact, Trump has contributed to Clinton's campaigns in the past. He has been very much a part of the political establishment. He's been running a counter-establishment campaign, but that doesn't necessarily make him anti-establishment. It just means that he wants to be perceived that way. And there hasn't been anyone with both the credibility and inclination to challenge that. Carson had the credibility, but he wasn't running a campaign based on criticizing other candidates. Bush had the inclination, but who would believe Bush complaining that Trump was too establishment.
The anti-establishment people in the media are generally pets of the establishment. The establishment media trots them out occasionally so as to laugh at the things they say. And if they do believe their own statements, they are so grateful that Trump is saying anti-establishment things that they ignore if he means them. It's only recently that Cruz has begun to pick on this. We'll see if it matters as the primaries continue.