I'd agree with you that the current admin
is anti-intellectual, but this speech is not a smoking gun.
For those who need spoonfed, here is the full speech: https://www.youtube.com/watch?v=0FR65Cifnhw
It's JD Vance's keynote speech at the 2021 National Conservatism conference. The speech, which I've just skimread, is mostly well-worn US conservative complaints about US higher education. He also talks about red-pilling because he's down with the kids, and he adds Jesus sprinkles in case you forgot he's Christian.
The speech is dull but it's bookended with two spicy statements, both of which you mostly quoted. The latter statement is not his words but a quote from Nixon.
Opening statement: «So much of what we want to accomplish, so much of what we want to do in this movement in this country, I think are fundamentally dependent on going through a set of very hostile institutions - specifically the universities which control the knowledge in our society, which control what we call truth and what we call falsity, that provides research that gives credibility to some of the most ridiculous ideas that exist in our country and so I'm excited to close this conference with this particular set of remarks, because I think if any of us want to do the things that we want to do for our country, and for the people who live in it, we have to honestly and aggressively attack the universities in this country.»
Closing statement: «I really want to end this on an inspirational note [...] and the person whose quote I ultimately had to land on was the great prophet and statesman Richard Milhous Nixon [...] there is a season for everything in this country and I think in this movement of National Conservatism, what we need more than inspiration is we need wisdom, and there is a wisdom in what Richard Nixon said approximately 40-50 years ago. He said, and I quote: "the professors are the enemy".»
EDIT: And for the context of the Nixon quote, it comes from a private conversation Nixon had with Henry Kissinger in the Oval Office on December 14, 1972, recordings of which were released in 2008: «Henry remember... we're gonna be around and outlive our enemies. And also, never forget, the press is the enemy. The press is the enemy. The press is the enemy. The establishment is the enemy. The professors are the enemy. The professors are the enemy. Write that on a blackboard 100 times and never forget it.». It's worth noting that Nixon was already keeping an "enemies list": https://en.wikipedia.org/wiki/Nixon%27s_Enemies_List