Tarantulas' Origins
Tarantulas' and the Tripredacus Council's origins in Beast Wars were unrevealed beyond them not being descended from the Autobots or the Decepticons, at least not in a way that they would be affected if Optimus Prime and Megatron I were killed in the past by a time traveler which would erase their descendants. There was vagueness, uncertainty, and fan inteprations.
The Unicron line, the writers' Cybertronian Empire idea, and my own alternate suggestion.
Unicron (Popular Idea)
We know that after Tarantulas' death, Megatron II called him a "Unicron's spawn". While a lot of people took it literally for some reason, both story editors, Bob Forward and Larry DiTillio, have said they never meant it that way (it was meant to be equivalent to "son of a bitch") and that Forward regrets the line.
He was depicted as a herald of Unicron in 3H Productions' Beast Wars comics and the Japanese "Yokohama Decepticon Secret Base" manga, and rumored to be a descendant of Unicron in Transformers Roleplaying Game, but all three are obviously different continuities from the Beast Wars cartoon and shouldn't be viewed as proof of an unicronian origin for the BW cartoon Tarantulas (also, the 3H continuity was only a BotCon comic anyway, basically Hasbro-condoned fanfiction).
(A 3H panel left, a "Yokohama" panel right. I couldn't find any 3H panels outside tfwiki, so no Unicron appearance or mention in that one)
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Cybertronian Empire (Would-be Canon?)
DiTillio, in the same message where he said that Forward regrets the "Unicron's spawn" line, said that they never settled on an origin for Tarantulas and the Tripredacus Council– but had the show continued, "It might very well have been that they were part of the Liege Maximo or Jhiaxus faction from the comic books".
Said faction is/was the Cybertronian Empire from Marvel's Generation 2 comic, a sequel to Marvel's G1 comic. It got started after the disappearance of Optimus' and Megatron's core groups when the Ark crashed on Earth and were gone for four million years – some Decepticons left on Cybertron recovered their ancestral ability to reproduce by mitosis, called "budding", which their creator Primus had locked away after the first ever transformers had used it to multiply faster to spread through the planet.
(Budding event on the left, semi-identical budded army on the right)
These Decepticons left Cybertron to start their own empire, led by Jhiaxus who was subordinate to Liege Maximo (one of those first ever transformers, and some kind of an ancestor to Megatron I, despite Megatron I being created long after budding had stopped and been forgotten). They then embarked on a campaign of galactic conquest, terraforming and genocide, which eventually led to them encountering the Autobots and Decepticons and going to war with both. The Cybertronian Empire didn't only consider all the other species inferior, but also viewed the other transformers (even the Decepticons) as primitive and disdained them. When the comic was cancelled because the sales weren't high enough for Marvel, the last issue saw the Autobots and Decepticons fighting against the Cybertronian Empire as allies and Jhiaxus died, but Liege Maximo was still alive on the last page and still had a massive army at his command, and presumably the ability to bud more of them as needed.
Also, they're evil because budding diluted Primus' life force in them, which made them amoral, emotionless, and lacking in individuality and which is why they wipe out other species with clinical detachment, unlike Decepticons who are more sadistic.
It makes sense, but there's a slight issue. If Tarantulas was an offspring of somebody in the Cybertronian Empire, he could of course have changed his appearance to blend in with the Predacons (Cybertronian Empire's members are all semi-identical, possibly because of the mitosis or lack of Primus essence), but does the above description of the average Cybertronian Empire bot's personality and the effects of generations of budding describe Tarantulas all that well? He's definitely not emotionless, and he's definitely sadistic. A Cybertronian Empire bot would also be able to just birth an army and wouldn't need to hotwire protoforms.
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Paradron (Secret Third Option)
My own suggestion, which I don't think has the plot hole the Cybertronian Empire origin would have.
In the G1 cartoon continuity, season 3 episode "Fight or Flee", Paradron is a planet settled by Autobots who fled Cybertron during the "Fourth Great War" which seems to have been before the war Megatron I started, and it's the only known cybertronian colony besides Antilla (where everybody died from cosmic rust an eternity ago). At this point it's important to note that in the G1 cartoon the labels "Autobot" and "Decepticon" predated the current war, because in the G1 cartoon the Autobots and the Decepticons were races; one created by the quintessons to be workers and consisting of "civilian" altmodes, and the other created by the quintessons to be combat drones and consisting of "military" altmodes. While there were civilian vehicle Decepticons who were built by the Decepticons themselves or not implied to be former Autobots, and there was one or two military vehicle Autobots who weren't implied to be former Decepticons, the civilian vehicle/military vehicle division was still the norm during Optimus Prime's and Megatron I's war and they kept mostly building their own type, and the paradronians were all civilian altmode types – and relevantly, every depiction of pre-beast Tarantulas is either a car or a motorcycle, which implies Autobot ancestry like paradronians all have.
The refugees lived on Paradron away from the war until they were discovered by the Decepticons and Paradron was occupied. The Autobots arrived to fight the Decepticons, but weren't able to retake it and so Rodimus Prime and Ultra Magnus decided to destroy the planet to deny the Decepticons its abundant energon. The Autobots then relocated the surviving Paradronians to Cybertron and Rodimus callously told their leader that Cybertron is better than Paradron was anyway.
It'd be insane if a lot of paradronians and some of their descendants built later on Cybertron weren't angry and vengeful towards both the Autobots and the Decepticons for enslaving them and then destroying their planet (propably killing many). So that's a motive.
And because paradronians left Cybertron before Optimus Prime's and Megatron I's war and only regained contact with Cybertronians after Unicron's defeat, Prime and Megatron I being killed in the past, their factions disintegrating, and their descendants being wiped from existence would not erase a paradronian or a paradronian-descended bot from history, because Paradronians had no relation to Prime's and Megatron I's groups. In fact, destroying Prime's and Megatron's / Galvatron's groups and descendants would be a win for paradronians because then Paradron wouldn't get destroyed. Also Paradron was entered through a weird space anomaly, which might mean it's located in some kind of a different universe and possibly out of Unicron's reach, in which case it wouldn't be paradronians' problem if no one was there to stop unicron in 2005 (or at least paradronian bots could think so).