From Wars of The Jews by Josephus, Book II Chapter XI:

...when the multitude were come together to Jerusalem, to the feast of unleavened bread, and a Roman cohort stood over the cloisters of the temple, (for they always were armed and kept guard at the festivals, to prevent any innovation which the multitude thus gathered together might make,) one of the soldiers pulled back his garment, and cowering down after an indecent manner, turned his breech to the Jews, and spake such words as you might expect upon such a posture. At this the whole multitude had indignation, and made a clamour to Cumanus [Oh yes, that's his name], that he could punish the soldier...

They riot, throwing stones at soldiers etc. Reinforcements show up, the crowd becomes a crush.

...the violence with which they crowded to get out was so great, that they trod upon each other, and squeezed one another, till ten thousand of them were killed....

I don't think that's conclusively a fart. And the real issue would have been the disrespect for their temple. So a fart killing that many is just a click baity titled, but not entirely wrong.

This incident happened under Cumanus. The only sources I can find for him are Tacitus and Josephus, with only Josephus mentioning it.

(This is my first top level comment ever, hopefully it's good enough.)

I can't comment on the story in Josephus, but the one in Herodotos is actually not quite what the author of that article made it out to be. Herodotos lets rip at 2.161-169.

In 570 BC, the Egyptian king Apries, at the urging of the Lybians whose lands were being usurped, sent a large army against the Greek settlers of Cyrene (near modern Benghazi). The Greeks, however, proved silent but deadly: contrary to everyone's expectations, the Egyptian army suffered a crushing defeat. Believing that Apries deliberately sent them out to die, the survivors banded together with the relatives of the fallen and rebelled against the king. Apries responded by sending a trusted man named Amasis to dissuade the rebels, but they instead proclaimed Amasis their true king. Amasis figured that this trumped his previous job description and decided to roll with it. He consequently prepared to march against Apries.

Undaunted by the backfiring of his plan, Apries tried again:

He sent against Amasis an esteemed Egyptian named Patarbemis, one of his own court, instructing him to take the rebel alive and bring him into his presence. When Patarbemis came and summoned Amasis, Amasis, who was on horseback, rose up and farted, telling the messenger to take that back to Apries.

-- Hdt. 2.162.3

When Apries heard of Amasis' refusal, he blew up and had the messenger Patarbemis mutilated. This outrage so infuriated the Egyptian population that whoever was still on Apries' side now turned coat. Apries, left with no other option, gathered his personal guard and all the mercenaries money could buy, and marched out to meet Amasis in the field.

In the hot wind of the Egyptian desert, the two armies came to blows - and Apries' outnumbered forces were swiftly overwhelmed. Apries was taken prisoner by Amasis. For a while he was kept alive, but the people demanded his death, and so Amasis' men took a rope and squeezed the air out of him.

You'll notice that at the time of the fabled fart, the rebellion was already well under way. Amasis passing gas was only his way of rejecting a summons to the royal court, which started a chain of events that worsened a situation that already looked pretty bleak for Apries to begin with. It seems somewhat disrespectful to blame the uprising on a fart when in fact it was triggered by the unnecessary death of thousands of Egyptians.

Kinda of an unrelated question, but did the ancient Mediterranean world have any vulgar vernacular (farts) or euphemisms for passing gas?

The bit about Josephus from the article is a little disingenuous here. As Josephus was quoted by /u/ReverendPoopyPants, there is no text to realistically say the Roman soldier farted, but we can conclude a few things from the passage:

1. The Roman soldier was stationed along the cloisters of the Temple of Jerusalem.
2. He exposed himself to the crowd "and spake such words as you might expect upon such a posture".
3. The crowd rioted.

The question should rather be rephrased as "why did the Jewish people riot when a Roman soldier bared himself?"

In first-century Palestine, the Temple of Jerusalem was considered the single most holy place for the Jewish cult. In contrast to the multitude of temples constructed to the many pagan gods by the Romans and performing daily sacrifices to the Roman emperor, a god king, the Jewish cult served only one god ("For I am God, and there is no other; I am God, and there is no one like Me" (Isaiah 46:9)) and worshiped him in one temple. There were synagogues then, but held a lot less significance compared to the Judaism we know today. The Temple Mount, where the Temple of Jerusalem was located, was considered one of the holiest sites in Judaism, and it was believed that this was the location where God's presence was the closest to earth. Jews all over Galilee would journey to the Temple, purchasing sacrifices like lambs or pigeons (depending on how much money you had), and taken through a series of gates and very elaborate rituals by a priest to sacrifice the animal and purify you of your sins. The Temple was an extremely sacred place to Jews at the time, up to the point of the Bar Kokhba/Second Jewish Revolt in 132 CE, where Rome utterly destroyed Jerusalem and the Temple.

At the point in time Josephus recorded this riot, Jerusalem was a province governed by Rome. In short, this caused a great deal of hostility between the Jews and Rome. Israel, to the Jews, was the land promised by God for "his people". There are various parts in Deuteronomy that say those who are not God's chosen should be driven out:

Because He loved your forefathers and chose their descendants after them, He brought you out of Egypt by His Presence and His great strength, to drive out before you nations greater and stronger than you and to bring you into their land to give it to you for your inheritance, as it is today." (Deuteronomy 4:37-38)

and

"It is not because of your righteousness or your integrity that you are going in to take possession of their land; but on account of the wickedness of these nations, the Lord your God will drive them out before you, to accomplish what He swore to your fathers, to Abraham, Isaac and Jacob." (Deuteronomy 9:5)

There are a great deal of other factors that contributed towards Jewish resentment of Roman occupation besides just religion, and I'll briefly list them here - high taxation, forced sacrifices to the Roman emperor, the decline of construction after Herod's death leading to Jewish unemployment, the high priests and Jewish nobility being bought and influenced by Rome, and a general belief that the Kingdom of God was nigh and would drive the non-believers out of their lands.

So think about it this way: You are a Jewish peasant, and made the journey to Jerusalem to purify yourself of your sins. You are celebrating the holiest day in your religion, Passover, where you commemorate Moses' liberation of your people from Egypt to deliver you here, to Jerusalem. You enjoy the celebration as much as you can while ignoring the row upon row of Roman soldiers looking down upon you from the walls of the Temple. They shouldn't even be there - the Temple of Jerusalem is a holy site for the Jews, and for a disbelieving pagan worshipping group to be granted access to this land chosen by God for you is an insult to you and your beliefs.

And then you see this Roman soldier, standing on land made holy by God himself, expose himself and shout profanities at you. How would you feel?

If you are interested in reading more about the historical environment of Jerusalem at the time, I highly recommend Reza Aslan's "Zealot: The Life and Times of Jesus of Nazareth". It's a historical study of Jesus and the time he lived in, and it's a fascinating read, especially if you're curious about the historical Jesus. All of what I wrote was based off his book.