This speech is part of a larger pattern of events between the Ministry and Dumbledore. Hermione has picked up on this context to correctly deduce it's meaning.
The context is vital to understanding what is going on here. Umbridges speech is made far more meaningful when remembering the circumstances in which it is being given.
Consider that for the previous few months Dumbledore has been trying to convince the world that Voldemort has returned, and that the Ministry has been actively opposing him. The world is comfortable, and any hint that the status quo is about to be breaking is regarded as distressing and unwelcome. After all, we see how the Minister of Magic responds when he's told Voldemort is back...
"It seems to me that you are all determined
to start a panic that will destabilize everything we have worked for
these last thirteen years!"
Harry couldn't believe what he was hearing. He had always thought of
Fudge as a kindly figure, a little blustering, a little pompous, but
essentially goodnatured. But now a short, angry wizard stood before
him, refusing, pointblank, to accept the prospect of disruption in
his comfortable and ordered world, to believe that Voldemort could
The conversation continues onwards with Dumbledore and the Minister actively opposing each other, leading to Fudge threatening the autonomy of Dumbledore and the school in general.
"If your determination to shut your eyes will carry you as far as
this, Cornelius," said Dumbledore, "we have reached a parting of the ways. You must act
as you see fit. And I...I shall act as I see fit."
Dumbledore's voice carried no hint of a threat; it sounded like a mere
statement, but Fudge bristled as though Dumbledore were advancing upon
him with a wand.
"Now, see here, Dumbledore," he said, waving a threatening finger.
"I've given you free rein, always. I've had a lot of respect for you.
I might not have agreed with some of your decisions, but I've kept
quiet. There aren't many who'd have let you hire werewolves, or keep
Hagrid, or decide what to teach your students without reference to the
Ministry. But if you're going to work against me ..."
The exchange ends with an indication that Fudge is going to clamp down on Dumbledore's running of Hogwarts.
"I will be in touch with you tomorrow, Dumbledore, to discuss the
running of this school. I must return to the Ministry."
Dumbledore, for his part, ignores Fudges threats and starts to spread the truth, starting with Hogwarts.
"Cedric Diggory was murdered by Lord Voldemort."
A panicked whisper swept the Great Hall. People were staring at
Dumbledore in disbelief, in horror. He looked perfectly calm as he
watched them mutter themselves into silence.
"The Ministry of Magic," Dumbledore continued, "does not wish me to
tell you this."
The campaign of information and attempts to rouse the public continues all summer, with the Ministry growing actively more frustrated and devious in their attempts to discredit Dumbledore and Harry and clamp down on the behaviour. Hermione - living at the headquarters of the Order and being friends with Harry (who tells them everything), is well aware of all this, explaining to Harry:
But you see what they’re doing? They want to turn you into someone
nobody will believe. Fudge is behind it, I’ll bet anything."
When Umbridge - a known Ministry toady - is appointed to Hogwarts, it's not a huge leap to assume she's there to continue these efforts. Note that Hermione already suspects this from the moment they spot Umbridge at the the feast, before the speech.
"She works for Fudge!" Hermione repeated, frowning. "What on earth’s
she doing here, then?"
Hermione scanned the staff table, her eyes narrowed. "No," she muttered, "no, surely not..."
And the actual content of the speech perfectly aligns with these suspicions - the content (as @TimB excellently put it in the comments) "reeks of a backward looking traditionalist fundamentalist worldview saying that new is bad, that traditions and by extension authority figures derived from those traditions is good".
In other words - the Ministry wants to keep things as they are, and so it emphasises the importance of "traditional values" and discourages change...
Hermione Granger: How about: "progress for progress's sake must be
...while casually threatening those who would seek to upset the existing system...
How about: "pruning wherever we find practices that ought to be
All of this - the rabble-rousing of Dumbledore, the attempts by the Ministry to control him and continue with "business as usual", the installation of a Ministry-appointed teacher at Hogwarts, and Umbridge interrupting Dumbledore to give her own pro-traditionalist speech at the feast clearly indicates one thing...
Hermione Granger: I'll tell you what it means. It means the Ministry's
interfering at Hogwarts.