First of all, I called the cause of the original tense ablaut the prefix -ā. Of course, that's only one theory. Another common one is the Laryngeal Theory (fun, Semitic gets one too)
In the modern Semitic languages, it is clear laryngeals like to mess up vowels a bit. They tend to drag vowels closer to /a/, because /a/ is the closest vowel shape-of-tongue-wise to most laryngeal consonants. For example, look at the Hebrew noun תפוח, originally tapuch, but the final /X/ caused the vowel to break, yielding modern tapúach. Note that this only happens at the end of a word: the plural is still regular, תפוחים tapuchim.
So, let's say instead that the past tense was formed by adding an ending (any one, it doesn't matter) to the present/future form. Let's just stick with -ā. So we've got (when adding in a new laryngeal-final root, *şīħ 'laugh')
Root - Past - Future
ptil - aptilā - aptil
mūt - amūtā - amūt
şīħ - aşīħā - aşīħ
Now, let's have a change like in Hebrew: the word-final laryngeal causes the preceding vowel to break and introduce an /a/. So, the last one becomes: şīħ - aşīħā - aşīaħ.
Now, we've got speakers who begin to analagize this introduced /a/ to all verbs, thinking that it's an integral part of the future tense. This vowel then overtakes the former vowel, leaving:
Root - Past - Future
ptil - aptilā - aptal
mūt - amūtā - amāt
şīħ - aşīħā - aşāħ
And then the final -ā marking the past tense drops through regular sound change, giving us our new ablaut system once again:
Root - Past - Future
ptil - aptil - aptal
mūt - amūt - amāt
şīħ - aşīħ - aşāħ
Alright, so we've got tenses and an adjective. But what about that mess of binyanim, where a verb can take a slightly diffeent vowel pattern to acquire a new meaning? How did those form? Let's look at a few specific patterns...
The Reflexive: Doing Something to Oneself
The Hebrew Reflexive Pattern: hitCaCeC (Third Person Singular Masculine Past Tense Reflexive)
The Arabic Reflexive Pattern: iCtaCiC (Imperative Reflexive)
The reflexive in most Semitic languages is made by inserting a "t" in between the first and second consonants, or, in the case of Hebrew, before the first. This likely came from an ancient reflexive pronoun, something like ta, which eventually prefixed itself to the verb. In most Semitic languages, it metathesized with the first consonant of the root, yielding their modern patterns.
In Hebrew, the metathesis wasn't as overarching as in Arabic. The hit- prefix remains in front of the root. However, if the first consonant of the root is /s/, /z/, /ts/, or /S/, the metathesis does occur: *S-K-L "look at, watch" becomes הסתכל histakel, instead of the usual *hitsakel.
The Intensive - Doing Something Intensely
Hebrew Intensive Pattern: CiCCeC (Third Person Masculine Singular Past Tense Intensive)
Akkadian Intensive Pattern: uCaCCaC (First Person Singular Future Tense Intensive).
As you can see, the intensive is usually formed by geminating the second consonant. In Modern Hebrew, however, the geminate has collapsed, making the vowels more important.
This was likely originally formed by reduplication, as many languages reduplicate a word to make it more intense.
(Conversely, modern Hebrew does the same to make an adjective less intense: ירוק yaroq "green", ירוקרוק yeroqroq "greenish". This is a form of diminutive.)
You don't have to reduplicate the whole word: just one syllable is fine. So, if I may use the root *D-B-R "talk" (which is now limited to the intensive pattern due to blurring of meaning)...
An older *daber could have been made intensive by reduplicating to *debdaber. Assimilation then changed this to *debbeber. Then syncope and assimilation again reduces this to *dibber.
The Causative - To Make Someone Do Something
Hebrew Causative Pattern: hiCCiC (Third Person Masculine Singular Past Tense Causative)
Akkadian Causative Pattern: ushaCCiC (ditto; the u- is a pronomial)
This is formed by adding sha-, or a reduced form ha-, to the root. There are two possible origins:
Like the Reflexive, this could have been its own word at one point, likely meaning something like "make" or "cause". The same thing happened in English, although it didn't expand that far: in Proto-Germanic, the verb "ian" (make) attached to the other of other verbs to make causatives: "fallian" (to cause to fall), "drankian" (to cause to drink) (Don't shoot me - I don't know Proto-Germanic, so I made up some of the forms... this change did actually happen, though).
This suffix caused the vowels to change, and then the suffix dropped. As a result, English is left with a few causative pairs, such as:
to fall ~ to fell
to drink ~ to drench
to sit ~ to set
to rise ~ to raise
However, this likely didn't happen in Semitic. Remember, I mentioned the sha- prefix earlier - it turned verbs into adjectives. So, we could have a cycle like:
*pil "lie" > *sha-pil "low-lying" > *shapil "to make low-lying" (ie, "to cause to lie down").
Hey, look! If we ignore the adjective in the middle, since speakers aren't going to remember the order of events generations after they happen, we have a causative: "lie" became "to cause to lie". So sha- could be reananlyzed as a causative.
The Passive - To Be Done
The passive originated from a prefixed n-, likely deriving from a verb meaning "to be" or "to become" followed by an adjective. In Hebrew, the basic passive pattern is niCCaC.
However, there is more than passive possible. I'm not exactly sure how these others formed... Posit your own hypotheses!
For example, Hebrew has two passive binyanim: A passive intensive and a passive causative:
Active | Passive
Intensive: CiCCeC, CuCCaC
Causative: hiCCiC, huCCaC
As you can see, the passive was formed by taking the active pattern and changing the vowels to u-a. How n- caused this, I'm not sure... Mabye the prefix was more complex at first?
Binyan Formation in Action - Or How Politics Affects Language
Well, that's fine 'n dandy. We've got several hypothetical examples. But can we actually trace any of them?
Yes, actually. Modern Hebrew colloquially has developed an eighth binyan, called hitpu'al, which is the passive of reflexive: "to be made to do something to oneself". It was created in the 1940s
This sounds a bit ridiculous at first, but it does have a use. For example, the root *P-T-R "set free" in the regular reflexive means "set oneself free", or by extension, "to resign". Now, it's perfectly likely to talk about someone being made to resign themselves...
Plus how about "to be made to wash oneself"? Or "to be made to volunteer oneself"? Those are relatively useful.
Anyway, it's origins. In 1948, an Israeli politician was forced to resign (you know how it goes...). Being very frustrated, he coined a new binyan in a speech expressing what happened, by analogy. Let's look at this:
Active | Passive
Intensive: piter "he set free (ie, he fired somebody)", putar "he was set free (ie, he was fired)"
Causative: hiftir "he caused somebody to be set free (ie, he caused somebody to be fired), huftar "he was cause... you get the point)
So, piter > putar, hiftir > huftar...
Now we have hitpater "he set himself free/resigned". Well, both of those passives are formed by using the vowel pattern u-a, so...
Hitputar! "He was made to set himself free"/"He was resigned"
Of course, this is nonstandard, but shows the powerful force of analogy.
And if you were curious, here's the root *P-T-R in each of the binyanim (all 3 Sg Masc Past), for meaning comparison:
Pa'al (Default): פטר patar "he dismissed, he exempted"
Nif'al (Simple Passive): נפטר niftar "he was dismissed, he was released, he passed away"
Pi'el (Intensive): פיטר piter "he fired, he discharged"
Pu'al (Passive of Intensive): פוטר putar "he was fired, he was discharged"
Hif'il (Causative): הפטיר hiftir "he caused smby to be fired"
Huf'al (Passive of Causative): הופטר huftar "he was caused to be fired by somebody"
Hitpa'el (Reflexive): התפטר hitpater "he resigned, he got rid of somebody"
*Hitpu'al (Passive of Reflexive): התפוטר hitputar "he was made to resign, he was gotten rid of"
Okay, that's it for now. From here it should be easy to see how this system can be expanded to include all sorts of concepts.