. . . the USA offered a place where everyone can do and say as they please, so long as they don't hurt each other. It's a great thing.
You are correct as a matter of law and generally where "hurt each other" is used in a directly measurable sense.
The point is that language can be (and has been) used to "keep those people in their place" for many groups that are "those people" and that are being "kept in their place". Those doing the keeping generally benefit while those kept generally don't, and most often suffer for it. That results in a serious kind of hurt, both economic and social, that's generally not illegal, but should nonetheless be condemned in the strongest possible terms.
Hurling racial epithets is legal. Decent society should not tolerate it. It's but one example of where unfettered expression is anything but a great thing. There are many more. Most of the people who whine about "political correctness" (and almost all of these are far-right conservatives) cannot give a measured, rational, and persuasive argument for their contentions. This was put quite well (and an excellent example as expressed by Donald Trump was offered) by a comment the on Washington Post:
It's no longer considered polite to put down women, gays, Hispanics or blacks for being who they are. To most of us, this is called manners, and it's what we were taught as children. To conservatives, this is called "political correctness".
What do you do when the core foundations of your beliefs can't be spoken openly? You can speak in code and dogwhistles, and there's plenty of that in conservative circles. You can say one thing to your conservative friends and another to everyone else, and there's even more of that going on.
But what you really want is to seize back the public space. So you don't have to dissemble any more. Because while conservatives accuse liberals of political correctness, it's not really the liberals who have to hide their deepest beliefs from the world - they're just saying what's on their mind, and it's already "PC". It's the conservatives who must bite their tongues.
Trump had them at "Mexicans are rapists", he's done nothing since to increase his appeal. He doesn't need to. He promises conservatives a world in which they can talk like their grandfathers did, and those whom it offends can just suck it up.