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?

Umbridge spoke for a long time, but only small snippets of it are recorded in the book. While it is theoretically possible that she said some stuff "off-screen" that gave away that the Ministry would be interfering, for the sake of answering this question let's assume that we have enough information in the parts that we saw.

Here are the snippets of her speech:

"The Ministry of Magic has always considered the
education of young witches and wizards to be of vital
importance. The rare gifts with which you were born
may come to nothing if not nurtured and honed by
careful instruction. The ancient skills unique to the
Wizarding community must be passed down through
the generations lest we lose them forever. The
treasure trove of magical knowledge amassed by our
ancestors must be guarded, replenished, and polished
by those who have been called to the noble profession
of teaching."

"Every headmaster and headmistress of Hogwarts has
brought something new to the weighty task of
governing this historic school, and that is as it should
be, for without progress there will be stagnation and
decay. There again, progress for progress’s sake must
be discouraged, for our tried and tested traditions
often require no tinkering. A balance, then, between
old and new, between permanence and change,
between tradition and innovation..."

"... because some changes will be for the better, while
others will come, in the fullness of time, to be
recognized as errors of judgment. Meanwhile, some
old habits will be retained, and rightly so, whereas
others, outmoded and outworn, must be abandoned.
Let us move forward, then, into a new era of
openness, effectiveness, and accountability, intent on
preserving what ought to be preserved, perfecting
what needs to be perfected, and pruning wherever we
find practices that ought to be prohibited."

I would argue that none of the concepts Umbridge discusses actually indicate Ministry interference. The first paragraph points out the unique mission of Hogwarts, the second paragraph points out that every headmaster changes things up a bit, and a certain balance is necessary, and the third paragraph says that we have to constantly look back and see if we are making things better or worse.

I don't think any of that is particularly controversial. In fact, I wouldn't be surprised if other headmasters made similar speeches upon starting at Hogwarts (though I'd expect some of them to at least have the sense to not give the speech to a room full of sleepy, inattentive students).

I think that the entire issue here is with the very first sentence:

The Ministry of Magic has always considered the education of young witches and wizards to be of vital importance.

Everyone agrees that education is important, and should be done properly. So why mention the Ministry here? I would argue that it is this insertion that is the indicator of Ministry interference. By telling us that the Ministry values education, she is saying that whatever follows (e.g. some things need to be reevaluated) will be Ministry-oriented. Had she left out the first sentence there would have been no indication of Ministry interference (though there might have been indication of her own interference).

In fact, we can make an observation based on the reactions of others besides Hermione. During the speech we find the following:

Professor McGonagall’s dark eyebrows had
contracted so that she looked positively hawklike, and
Harry distinctly saw her exchange a significant glance
with Professor Sprout as Umbridge gave another little
"Hem, hem" and went on with her speech.

Yet, interestingly, this does not occur after the second or third paragraph, which is where Umbridge actually lays out what the "interference" would consist of. Instead, it occurs after the first paragraph when all Umbridge said is that the Ministry values education. It would seem, then, that McGonagall and Sprout already realized something was up, just from the first paragraph. If so, it was probably from the mention of the Ministry, because as noted above that is entirely superfluous unless the Ministry will be interfering.

Therefore, I would assume that Hermione also figured out right from the start that Umbridge would be initiating Ministry interference. The specific examples mentioned by Hermione (and the other answers here) are just that — examples of how the Ministry will be interfering once we already know that they will be 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, and who are "we"?

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".

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 practices that ought 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 the Ministry of Magic is interfering.

Dumbledore was the headmaster of Hogwarts, whereas Umbridge was just an underling working under the direction of the headmaster. For Umbridge to publicly pass absolute judgements against the policies or philosophies of Hogwarts was not a constructive action but therefore a hostile one. Dolores was able to get away with inappropriate and socially out-of-place comments like that because she had the protection of the Ministry to fall back on, hence the reason for Hermione's comment. It doesn't take a genius to see that, all you have to do is pay attention to what is going on around you, as Hermione often did.