This isn't askscience. but if I may go on a non-historical fact spree (in addition to any interesting historical answers that me be written), I'm familiar with throat-singing and practice it myself from a western, extended singing techniques point of view. Why would it be interesting with a technical runthrough in a historical context? Because the techniques throat-singing is based on are universal! It stands to reason that different types of tradition can emerge independently over the globe, and should because the techniques are often used in everyday speech. Throat singing can often have a very mysterious and exotic quality for a western audience, and I hope dispelling this fancy can help view different singing traditions and techniques from a more impartial viewpoint.
Throat singing refers to a a throng of techniques and styles from different cultures, the word is often misused in a western context instead of overtone singing, and vice versa.
Tuva consists of two different techniques compared to "normal singing" (there's really no such thing), the first is using your vestibular fold together with your vocal folds (in Swedish we actually call the vestibular folds the "false vocal folds"). Now, in all manners of singing the vestibular folds are used in some ways (in any case as something you should be wary of in your body, even though you keep them still), but they are more pronounced in styles such as tuvan throat-singing, or (metal) growling. In growling you let the folds vibrate more chaotically, and your normal vocal folds are used less than in speech or common singing. This is of course a simplified version, since growling is a very broad field where some have more noisy output and some have more tonal output. In throat singing, you do the same but "lock" the frequencies of both set of folds to a tonal relationship that is in the harmonic (overtone) series.
Different styles of throat-singing use different overtones, and produces different sounds. The fun part is that the different folds interact, because they use the same stream of air from your lungs. I have yet to find a source on exactly what happens physically, but I would venture a guess that the vestibular folds affect the amplitude of the frequency from the voal folds, creating an effect similar to amplitude modulation:
https://youtu.be/VHZ0KOvRj-c?t=41s
In this video, one oscillator modulates the amplitude (often the word "volume" is misued in a daily context instead of amplitude) of another oscillator, going from below audio frequencies (below what the human ear would think of as a sound) and into audio range. The situation with the different sets of folds is way more complex than that, though, but hopefully it can help someone reading this how to conceptualize and visualize throat-singing as an audio effect.
At 1:29, directly after the mark in this link, the legendary Ella Fitzgerald does shortly use her vestibular folds in a light fashion, but you can still hear it is the same kind of sound as throat singing:
https://youtu.be/OFclKdniaDk?t=1m27s
Now, the next technique I'm going to run through is the overtone singing part of this. Without going too far into the complex math, all (complex) sounds can be said to be constructed with a set of sine waves, produces a spectrum of frequencies. Our throat produces a tone that is very rich in harmonic content, meaning that it needs to be constructed with a damn good amount of sine waves. We use our mouth, our lips and tongue, to mainly filter away harmonic content. The more closed your mouth is, the less harmonic content and vice versa. When people talk about a wah-wah pedal for guitars, they call it that because it is in a sense doing the same effect as your mouth does when saying "wah". A wah wah pedal is a low pass filter (low pass filter passes low frequencies, and cuts high frequencies, compared to a high pass filter) that you control with your foot. But your mouth can do a bit more than that, it can also boost certain harmonic content. For example, an "a" is close to how your throat sounds without your mouth, but an "e" uses your tongue and lips to boost the upper frequencies of the spectrum. Different languages have their own vowel sounds, emphasizing different parts of the spectra, even though we write them with the same characters.
Overtone singing is about boosting a single sine wave in this spectrum until it becomes so strong that it pops out and sounds like a sound of its own to our brain. The effect is found all over the globe, in both music and language.
You can try out this effect in action on this page, where I created a patch for the occasion:
The second panel called VCF (filter), with the cables inserted, is what you are looking for. Turn the knob called freq (frequency) and you'll hear the wah effect. Now, turn up the knob called reso (resonance, a synth term for feedback in filters) and again turn the frequency knob and you'll hear "throat singing".