I've seen this one used far too many times to ignore it, so let me have my take on some lazy thinking mixed in with some despicable oppression.
In a nice, clean, sanitised form, the tone "argument" (I would rather not dignify it any longer, thanks) is the idea - the bad idea, I should point out - that the tone a statement is made in somehow affects the accuracy of the statement. To put it rather less generally, if I swear, shout or generally do anything the other person finds offensive, this somehow makes my statement inaccurate or wrong. To show how little sense this makes:
Prima: "The sun goes round the earth."
Secunda: "No it doesn't, the earth goes round the sun!"
Prima: "How can you say that? It's the sun that moves in the sky."
Secunda: "Yes, from our position it looks like the sun's moving, but it's actually the earth."
Prima: "I don't believe you. I don't trust science."
Secunda: "How fucking hard is it? The earth goes round the fucking sun! We have proof!"
Prima: "You know, maybe you'd win more people round to your side if you were nicer about it."
Now do you see the problem? Prima utterly ignores Secunda to the point where Secunda gets a little angry, then rejects Secunda and Secunda's statement for swearing and "not being nice". I am only lucky that I used an example as obvious and accepted as the earth going round the sun - and even then, I think someone might use the tone argument on me.
An issue with this is that the style of my statement does not affect the content of my statement. It's the equivalent of said statement being rejected as false for being written in, say, red, or sprayed in pink across a bridge. It makes no sense whatsoever. It also really doesn't matter whether I spell "The earth goes round the sun" out in fireworks or whether I explain it at a lecture - my statement is still true. (OK, I concede, roughly true.)
I'll move off this nice, neutral, sanitised version and into its actual application: the oppression of the less privileged and the derailing of arguments. If a person with less privilege (woman, POC, etc.) tries to argue their case to a person with more privilege (man, white, rich, cis, etc.), they might end up getting angry at how their points aren't getting through. It happens. And if they do get angry - enter the tone argument! "You're angry, that means you can't be rational! You're not thinking this through properly! And besides, my research shows..."
...True logic and rationality are not found in a person's tone of voice. They are found in the substance of a person's words.