True, English doesn't help here. I can see how that may pose a problem when you come from English as your native language (and possibly others which have no case system to speak of).

I think the explanation given on the site is pretty clear. English doesn't have to worry about cases, and the example given is We talked about what you were reading. "What" acts like a hinge combining the two halves of the sentence because it both refers to "about" and to "reading": What were you talking about? What were you reading?

As "what" works both ways one might presume, coming from English, that the same mechanism works in Russian, but it doesn't. You can translate the first half of the sentence like this: Мы говорили о том, which in English is rendered as "we were talking about what". But о том may not be the object of the verb читать here, so you cannot just go on saying о том вы читали as you can in English. Читать requires an object in accusative case. Therefore a comma separates the two phrases neatly, and a second pronoun is inserted after it which fulfils the requirement of accusative case: Мы говорили о том, что вы читали. This is more like "We were talking about that / which you were reading". You might feel like this is a slightly more convoluted way of saying the same thing as with "what", and in fact I have found that "Could I express this in a slightly more convoluted fashion?" is a thought which helps me write better Russian, even coming from German as a native language. Russian likes neatly separated complete main and dependent clauses, where English tends to condensate them.

It's a bit of an oversight that the website you linked to calls it T-K construction, saying that a second pronoun beginning with "к" usually follows, and then goes on primarily giving examples with что. I don't think there's a standard name for the phenomenon as it would be something a native speaker never needs to think about.