I'm aware that words can be cheap sometimes, that making promises or identifying problems is easy while fulfilling those promises or fixing those problems is a lot harder. Even so, I still can't really agree that there's any such thing as "just" words.
Words have a lot of power. While they're not equivalent to actions, they are the precursor to actions; it is difficult to fix a problem without first identifying it, after all. And while words are formed from thoughts, they form the framework with which we are even capable of defining a thought, and have their own influence on the way people think as a result. Words can help and support others, or deal them a great deal of real and significant harm. Every word we use is probably going to have some kind of impact.
And the power that any given word has will change relative to the people involved. Every person has their own context for a word. Sure, most people share a lot of the same context for a lot of words. But words like racial slurs, for example, have a very different impact on members of a minority class than they will on people who don't share that history of oppression.
I may not be a linguist or a psychologist, but I'm reasonably certain that nothing I've laid out here is all that controversial. But then, what about that phrase "they're just words"?
It is most often used as a dismissive phrase, one that rebukes others to not let those words in question affect them. Effectively, it denies that words usually have an impact, especially words that usually can seriously hurt someone. On top of that, it asserts that it's possible to choose whether or not a given word can affect you, usually without considering how much greater any given word's impact might be on someone other than the speaker.
Worst yet, it is an abdication of responsibility; the longer version of the phrase is "it's not my responsibility to think about the words I'm using, it's your responsibility not to let those words affect you." Or, more bluntly, "it's not my responsibility to avoid hurting you, it's your responsibility to avoid being hurt." In that sense, it is a phrase that is all but fundamentally hostile to civil discourse or the concept of responsibility for one's
It is a phrase I despise, and one that causes me to wince when I see it used. It is a phrase that I believe should not ever be used.