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At the start-of-the-term dinner, Dolores Umbridge interrupts Albus Dumbledore and gives a lengthy and quite strange speech. As summarized in the book,

Hermione Granger: There was some important stuff hidden in the waffle.

Ron Weasley: Was there?

Hermione Granger: How about: "progress for progress's sake must be discouraged"? How about: "pruning wherever we find practices that ought to be prohibited"?

Ron Weasley: Well, what does that mean?

Hermione Granger: I'll tell you what it means. It means the Ministry's interfering at Hogwarts.

Of course Miss Granger is a genius and I am not, but I do not see any way to infer "the Ministry's interfering at Hogwarts" from what Umbridge actually said.

Could you help explain how this was inferred? Is the speech supposed to mean something like "the Ministry's interfering at Hogwarts" to a normal reader, or is it just a Sherlock Holmes-like move to show us how clever Hermione is?

P. S. English is not native to me, so please feel free to correct the question!

To understand this you have to appreciate that the Harry Potter series of books were not written in a vacuum, and that JK Rowling is a politically active person. The idea of the British Government interfering in education has been an issue here for a couple of decades and Rowling's works mirror the real world situation in the UK in her Wizarding world.

The words of Dolores Umbridge reflect the typical doublespeak that politicians use in the real word. "Progress for progress's sake should be discouraged" means that you should not do something progressive just because you can, in our world it would be something like the right of gay marriage.

"Pruning wherever we find practises that out to be prohibited" means stopping the schools from teaching subjects that the Ministry did not approve of such as muggle studies, in the real world in Britain subjects such as music teaching and media studies had reduced funding.

The Ministry is not allowing the staff at Hogwarts to make its own policies about what it should teach and how it should teach them. That's how it is interfering.

"progress for progress's sake must be discouraged"

Who decides whether an innovation is a good thing or just "progress for progress's sake"?

"pruning wherever we find practices that ought to be prohibited"?

Which practices are those?

Both of these implicitly assume a judgement on what should be discouraged or prohibited. Since Umbridge was appointed by the Ministry it follows that it is the Ministry judgement that will be applied. Presumably in the past the Headmaster of Hogwarts was free to run the school as he saw fit, answerable only to the Board of Governors. Umbridge's speech was a coded announcement that in the future the Ministry would be setting policy directly, which is what Hermione described as "interfering".