Why is the definite article used in: ”What happened to the rest of "the" soup”?

Your confusion is coming from determining what counts as the soup. Soup is something created in quantity, so there is often the case of some of it being left over. When people ask about the rest of something, they are referring to what was left over, that is, unconsumed and still existing. That the question is being asked means that someone moved the leftover soup somewhere unknown. You need the here because you were talking about a very particular leftover soup, and the adjective phrase underscores that point.

Indeed, the soup that we ate yesterday is apparently some, not all, of the whole soup prepared

This part is wrong. In the context of the sentence that you mention, it’s most likely that “the soup that we ate yesterday” refers to all of the soup that was prepared, even though not all of it was eaten.

The title of your question suggests it's about why the definite article is used, and the first part of your question why a definite article as opposed to an indefinite article (or no article). In terms of this particular sentence, it needs a definite article because it's referring to a specific soup—that is, the soup we ate (a portion of) yesterday.

At first glance this appears to perhaps be more generally related to the phrase the rest of X. The phrase seems to lend itself to objects which, if taking an article, would use the definite one. But exceptions readily come to mind: for example, "What happens to the rest of an afternoon [once you've done xyz with it]?" (Incidentally, it's difficult to think of a context in which one would use an indefinite article after the past tense happened to... perhaps that's also why the example sentence demands a definite article? But that's probably not the point the author of the Learner's Guide was making.)

But your more urgent point seems to be not so much the in/definiteness of the article, but rather what noun phrase is properly the object of the question. And rightly so, since, as you point out, the sentence could be parsed in a way that makes the question border on the absurd.

If we start with the basic form of the question, "What happened to X?" X is made up of three elements: the rest [of], the soup, and that we ate yesterday. One way to parse this sentence is with the soup as the main noun,† which we'll call β. So you start with β (the soup) and modify it with the other two—which you can do in either order, interestingly. The soup can become the rest of the soup or the soup that was eaten yesterday, but either way when you modify it with both you end up with "the rest of β that was eaten yesterday."

In other words, if the soup is the main noun of the sentence, the question could be rephrased as something like: "What happened to the remainder of that portion of soup we ate [yesterday]?" Or, as you put it, 'What happened to the (rest of the) soup that's now in our stomachs?" To which the answer might be, "Ummm...well, there isn't any left to speak of, if you're asking about the portion we already ate." Parsed this way, the question is logically its own answer: there was some soup; yesterday we ate a portion of it. What happened to the portion we ate? We ate it—what is there to inquire about? There is no remainder of the (portion of) soup that we ate yesterday. Or to go back to the β formulation: there can't be a remainder of some thing, β, that was eaten yesterday.

Of course, your interlocutor would assume you weren't asking this version of the question since you're likely attempting to engage in meaningful interaction.‡ Given the assumption that the purpose of language is meaningful exchange, the main noun of the sentence must be the rest. (It's mind boggling to think that this calculation is something we do as interlocutors almost instantaneously, yet attempting to reason out and describe why the calculation is logical has taken me hours of pondering). In this case, of the soup modifies the rest and that we ate yesterday modifies the soup. So in the basic form "What happened to X?" X is the rest, so that what we're asking is what happened to the remainder [of the soup—that soup we ate yesterday]?

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†I believe this is referred to in functional linguistics as the head of the nominal group, but I'm not a linguist and I could be anywhere from slightly wrong to completely off base.

‡This is reminiscent of Grice's Maxims. None of the maxims is strictly applicable here, but underlying all of them is the rationale that communication is a cooperative effort aimed at meaningful exchange. Thus, assuming we're not characters in a Samuel Beckett play, we presume our interlocutors aren't asking questions which are, on their face, absurd.

This is actually pretty easy to assess the answer. Here the answer is answer of this question which is known to us. So we now know when we use the.

The soup means not any soup. We are not talking about any soup rather than the one that we think of it immediately. So you and your friend go to a restaurant and the day after that he/she says "the restaurant was great". You just think of that restaurant you went yesterday because of THE.

Now the question itself, if they hadn't eaten soup, he/she wouldn't have said such sentence.

Always remember a man that flies in the sky is hero

1. man could be any one and he is not known
2. sky is the one sky we all know
3. hero is in general not any hero or the hero, just hero.

This should help you distinguish these three form of presenting a noun.

Having been encouraged to re-post my comment (which has no references I'm afraid) as an answer here it is:

You have provided the correct answer to this yourself when you wrote "There was soup. We ate some of it and left the rest over. What happened to the rest of the (whole) soup" the idea is that a batch of soup was made and we ate (or drank or had) some of it. The question is then about the whole batch.

Perhaps it makes more sense when talking about solids, for example "What happened to the rest of the turkey we ate yesterday?" There is usually leftover turkey, they're huge.

Having said that I think 'had' is more common than 'ate' in this context and avoids the logical conundrum.

You are trying to apply the same logic to two different propositions—eat soup and spend money.

The verb eat operates in a wide range of interpretations. We ate the soup could approximate, for example:

We tasted the soup. eat or drink especially in small quantities
We partook of the soup. have a portion (as of food or drink)
We had the soup. partake of
We ingested the soup. take in for or as if for digestion
We swallowed the soup. take through the mouth and esophagus into the stomach
We finished the soup. consume or get through the final amount or portion of ... food or drink

We can divide the soup. Still: One soup, many bowls. Not more soups.

When we ask Where’s the rest of the soup?, we assume that this soup was not entirely consumed—that we partook of it, that we didn’t finish it.

Therefore, when we ask Where’s the rest of the soup that we ate yesterday?, we can logically conclude that we are talking about the one soup of which we partook yesterday; as you noted, it would be absurd to inquire about the balance of a soup finished.

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The verb spend, on the other hand, operates in a narrow range; it’s all-consuming:

We spent the money. use or give out the whole of; exhaust
We used up the money. to expend or consume by putting to use—often used with up

When we ask Where’s the money we spent yesterday?, the answer is: Duh, it’s where we spent it—the store, the bar, the racetrack . . .

The question becomes more absurd when we add the rest of; what could be the balance of money used up?

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As for the use of the definite article . . . it signals that we are talking about something particular:

Where’s the rest of the soup [that I’m thinking of and assume you know what I’m talking about]?

           The rest of which soup? [We have leftover pea soup, tomato soup, miso soup . . .]

The pea soup—you know—the soup that we feasted on [ate large quantities of] yesterday.

           There’s not much of the pea soup left, but it’s in the refrigerator behind the orange juice.

What happened to the rest of the soup that we ate yesterday?

The is correct because “soup” is defined for/specified to the listener.

The is a determiner - basically a demonstrative adjective - the noun means "that particular noun of which we (speaker and listener) are [now] aware.”

(The is similar to “that” - in fact it is a form of the Old English “that” and often “that” can be used in place of the.)

The is used

(i) where the noun is well known to everyone:

"The moon is bright"; "The Kremlin is in Moscow"; (Everyone is aware of the the moon and the Kremlin.)

(ii) This applies also to the use of “the” to identify a class of nouns that the listener is expected to know:

"The dog (= any creature known as "a dog") makes a good pet."

"The harp (= any musical instrument known as "a harp") is difficult to learn."

"The oak lives for 600 years (= any tree known as "an oak");

“The car was an incredibly good invention.” (= any mode of transport known as "a car.")

(iii) where the noun has been already mentioned so that the speaker and listener are aware of the exact thing that is being discussed:

"I saw a (= a random example of a) cat in my garden this morning. The cat (speaker and listener know which cat is being discussed) caught a bird but the bird escaped."

(iv) where the noun is defined or described so that the listener will know what sort of noun it is:

"I saw the man that I met in Paris." that I met in Paris defines "man" and the listener now knows which man you are talking about.

"The decision to shoot the prisoners was made." "to shoot the prisoners" defines "decision" and the listener now knows which decision you are talking about.

(v) where the noun is later/retrospectively defined or described so that the listener realises which particular noun was being referred to:

"The cat stretched its legs and slowly walked across the room. It had lived in a small box below the window since John had rescued from the river”: lived in a small box below the window since John had rescued from the river makes the listener/reader aware of exactly which cat this is. You therefore use “the”.

(vi) to modify an uncountable noun by confirming the identity of that noun:

“I have information that you need” – no article = “I have [some / an example of / examples of] information of which you are unaware but which you need.”

“I have the information that you need” – article -> “I have that information that both you and I are aware that you need.” (You and I are aware because either (i) you have asked us for it, or (ii) I mentioned earlier.)

In both of these examples, you will see that “that you need” defines “information”, so you might think that “the” could be used in both. But it cannot be used in the first because only I am aware of the information – and in order to use “the” we must both be aware.

(vii) to modify a qualified uncountable noun by confirming the identity of that noun:

“I have {information about the spare parts} that you need” – no article -> “I have [some / an example of / examples of] information of which you are unaware but which you need.”

“I have {the information about the spare parts}” – article -> “I have that information about the spare parts that both you and I are aware that you need.” (You and I are aware because either (i) you have asked us for it, or (ii) I mentioned earlier.)