First, and this may seem trivial but it's quite important, we tend to think that when other people disagree with us, they're wrong and we're right. That's perfectly all right and logically necessary, but the problem is that we often have a very hard time considering the alternative (that they're right and we're wrong) even as an abstract possibility. It's often much easier to imagine that our opponent is dishonest or evil or stupid than it is to acknowledge that we might just be mistaken.
This is so for several reasons, but a subtly important one is this: simple soundbite explanations are almost always much easier to absorb and repeat than complex, nuanced explanations. Identifying the cognitive error we or our opponent may have made in good faith in formulating an opinion takes a lot more effort than simply concluding that they're dishonest or evil or stupid.
Second, what we hear about a situation is always heavily informed by whom we hear it from. If I tell you I was in an argument with Jack, and he used some false statistics to try to support his point, you're not getting Jack's side of the story, and unless you're very conscientious in trying to keep an open mind (or you're already very skeptical of me), you'll tend to form your perception of the situation based more on my view than Jack's.
Third is a phenomenon I've seen countless times in conversations and debates in many different topics: someone says something, and someone else says they shouldn't say that because it's wrong or it's false or it's offensive, and the first person protests that they have every right to say that because free speech. Well, yes, of course they have the right to speak their mind, but that's a completely irrelevant argument (and a pretty poor way to defend one's position). Freedom of speech necessarily includes the right to opine on the merits of the content of someone else's free speech, and criticism is not censorship. Indeed, it's almost antithetical to censorship; to engage with disagreement and articulate reasons for that disagreement depends upon admitting the contested idea into the the discourse, if only to be considered and rejected.
Fourth, any claim you hear more than once tends to sound more credible the more you hear it, and the more people you hear it from. And whether you consciously accept the claim or not, it still primes you to interpret new experiences in light of its paradigm. A lot of prejudice gains traction this way in a vicious circle of subjectivity. An initially not-at-all prejudiced cop, for example, might hear it said that black people commit more crimes, and vigorously reject the idea, but from that moment on black suspects will be more memorable as data points, and after a while he may find that he's pulling over black motorists at a higher rate than white ones, thus making proportionately more arrests of blacks, and eventually concluding by golly, they do form a higher percentage of the crimes he catches.
I should stress that this not in itself pathological; it's actually how correct ideas propagate, as well. But it's imperfect and can serve to amplify ideas regardless of whether or not they are true.
Fifth, and related to (4), is that once something becomes A Thing, a perceived general trend rather than a collection of individual instances, it becomes much, much harder to dislodge. It's like the Gish Gallop I talked about in this post; refuting a single example instance, even most or all actual examples cited, cannot prove that all those nebulous others (of which the refuted examples are just one picked at random) don't actually exist. "Okay, sure, that might be a bad example, but what about all those other times that A was B?" In a way, it's a manifestation of the principle I mentioned at the end of (1): a broad generalization is much easier to remember and work with than analyzing individual instances.
So there's the groundwork. What does it mean for the proposition that Political Correctness will be the death of free speech?
There's little doubt that some social norms have been shifting over time, and some things that were once accepted as normal (and normative) are now condemned. Making a racist or sexist or homophobic joke that once would have got laughs (or a polite but maybe uncomfortable smile) will now often attract open criticism. But these changes take time, and they take longer when the underlying idea is not obvious. The fact that jokes are jokes introduces some complexity that makes it easier to resist criticism, for example; it's quite natural and quite reasonable to disavow actual racist beliefs by insisting that you were really just telling a joke that you thought was funny. It can be quite difficult to understand, in some cases, why an apparently harmless joke is based upon (and reinforces) a fundamentally harmful assumption, and it's much easier cognitively to say "it's just a joke".
And so when you get a critical reply to what was meant to be funny, when you're treated like the bad guy for trying to inject a little levity, it's really hard not to conclude that they misunderstood and are wrong (too sensitive, too angry, to hateful) to be offended (1). And when they object, you might (3) defend yourself by saying you have a right to tell whatever joke you want, because free speech. And maybe you'll go tell a friend about what happened, who will of course hear it from your perspective (2). And since you've chosen them as a sympathetic ear, it's likely they'll relate it to a similar experience of their own, and now you have two instances of A Thing (4). And all it takes at this point is for someone to name the Thing, and then the label of "Political Correctness" catches on and examples suddenly appear everywhere you look.
There is no shortage of people willing to provide examples, many of which will be apocryphal or distorted (2). You will hear second-hand accounts of how this or that Act of Parliament or court decision criminalizes free speech, and because it fits into The Thing, you'll accept them as credible or even authoritative. Last year, for example, the House of Commons passed motion M-103, which condemns religious discrimination and Islamophobia in particular. It's a motion, and not a bill, which means it doesn't enact any kind of legal consequence; it's no more binding than Parliament voting to wish someone a happy birthday. But I keep hearing from people who think it means you can't criticize Islam or you'll go to jail. And even if this example of political correctness killing free speech turns out to be wrong, what about all those other examples(4)?
This doesn't mean, of course, that there isn't such a thing as political correctness, or that it won't be the death of free speech. I don't think that's the case, but for now, I just want to raise the possibility that however solidly real the phenomenon might appear to you, it could appear that way for reasons that have little to do with whether or not it actually exists.