Reading those words of his today we can easily enough agree with his assessment. But we have the benefit of knowing a lot of the details of Nazi atrocities that happened only after those words were spoken (the Holocaust; the SS on the Eastern Front; the Eugenics programs etc).
Was Churchill just giving a rhetorical flourish in wartime or did he really see the evil at the heart of the regime in a way that many around him had not woken up to?
It's partly rhetoric — that quote is taken from his very first speech to the Commons after becoming prime minister in May 1940, and so it's a significant moment for him as a statesman; to all intents and purposes, it's his mission statement.
But I do also think there was a contemporary sense that the Nazi regime is different. I don't necessarily agree that Churchill saw something that "many around him had not woken up to"; there were plenty of people who had watched seven years of repression in Germany with horror.
Were people anticipating genocide on the scale of the Holocaust? Maybe not. But they saw the Nuremberg Race Laws introduced; they saw Kristallnacht; they saw widespread political repression and the dismantling of German democracy in favour of a totalitarian regime. By the end of the 1930s, Germany was, if not quite a 'pariah state', then certainly a state viewed with mistrust and no small contempt by the traditional European powers.
The problem was that it was also an increasingly militarily powerful and expansionist state which didn't conduct its diplomacy in the traditional way, and wasn't responsive to traditional geopolitical pressures.
Even Chamberlain expressed moral objections to what was happening in Germany. This is taken from a letter he wrote to his sister after Kristallnacht:
I am horrified by the German behaviour to the Jews. [...] It is clear that Nazi hatred will stick at nothing to find a pretext for their barbarities.
In another letter, to a different sister:
No doubt Jews aren't a lovable people; I don't care about them myself; but that is not sufficient to explain the pogrom.
Source: Louise London, Whitehall and the Jews, 1933-1948 (2003), p. 106.
It wasn't because Chamberlain liked Hitler's regime that he pursued appeasement — rather, he wanted to forestall war for as long as possible.
All that being said, and in the context of May 1940, I don't think it's necessarily surprising to find a view in Britain of Nazi Germany as 'evil'.
Also, here's a slightly longer version of the quote, just for slightly clearer context:
You ask, what is our policy? I can say: It is to wage war, by sea, land and air, with all our might and with all the strength that God can give us; to wage war against a monstrous tyranny, never surpassed in the dark, lamentable catalogue of human crime. That is our policy.
Around 1931-1933 Churchill admits "I had no national prejudices against Hitler at this time. I knew little of his doctrine or records and nothing of his character" Winston goes on to say "I admire men who stand up for their country in defeat, even though I am on the other side." He did, however, want to ask someone close to Hitler "Why is your chief so violent about the Jews." So during the early 1930's Churchill did not really see the evil yet, but he did realize what was going on
Hope this answers your question or helps. I suggest reading Memoirs of the Second World War by Winston Churchill. It's very interesting and provides a unique (to me) view on what was going on before the war and how Churchill reacted to it.
All quotes are coming from the book Memoirs of the Second World War by Winston Churchill.