How did Tunisia's revolution (and Arab Spring) begin?

Its a truism, at least in the Lockian conception of government that any system of government that doesn't allow the governed a legal way to remove leaders that have become unacceptable (for whatever reason), in the long run guarantees itself extra-legal government change. All things end, and if there's no peaceful way to end a government, its end will not be peaceful.

So now lets get to the History of the matter...

Ben Ali for his entire tenure carried out rigged elections, with implausibly high winning percentages indicative of an electorate that didn't feel they had a real choice. His lowest reported victory was nearly 90%.

Meanwhile up until 2011 he was presiding over a country that was becoming increasingly young, literate, tech-savvy, corrupt, and unemployed. There was in fact a tremendous "generation gap", like 1960's in the US with the volume turned up to 11 (or even 12). A country like that, with no legal means provided for changing the dire system for its young adults, is of course a tinderbox^*.

One of the ways Ben Ali kept a lid on the "other methods" of getting rid of him, was by convincing Tunisians that he had US backing.

So the spark that lit the tinderbox happened on December 17, 2010. Mohamed Bouazizi was a 26 year old father of 6 who was forced to support his family by selling fruit on the streets. Officially, this didn't require a license, but unofficially you had to pay local officials for a "license". He couldn't afford that, so he suffered years of harassment and confiscations. As you can see from the stats I gave above, this situation was so average there, it practically made him a Tunisian "everyman".

On this particular day, he got another confiscation of his produce and equipment, but along with a public humiliation by the older (quasi?) police woman, and just got fed up with the injustice of it. He marched down to governor's office and demanded his property back. When the governor refused to even see him, Bouazizi got a can of gasoline, yelled "How do you expect me to make a living?", and set himself on fire in the street.

Now you're right that this alone might not be enough every time to produce an uprising. You need the tinderbox (which we had), the spark (Bouazizi) and if you'll permit the metaphor extension, you also need oxygen.

Word of this got out among the young of Tunisia, thanks to their internet connectivity. While Bouazizi was lingering in the hospital, protests organized on social media started to spread.

While this was going on, Wikileaks coincidentally released a huge cache of classified information taken from US embassies, which happened to include the one in Tunisia. Along with a dizzying array of corruption (surely only news to Tunisians in the details of it), young Tunisians were able to see that the US State Department did not completely back Ben Ali, but rather viewed him as a bit of a buffoon, and a growing problem. Not to mention some of the details of corruption in those cables were rather lurid, and perfect fodder in the new clickbait world of social media.

This same day Ben Ali felt the need to visit Bouazizi in the hospital as a conciliatory gesture, but it was too little, too late. Tunisians no longer wanted him and no longer feared him. On January 4th, Bouazizi died. Two days later every lawyer in the country went out on strike. The next day the teachers joined. That Friday, Ben Ali fled the country. The next day, he resigned.

For a good data-driven overview of what's been going on socially and politically the last decade or so in the Arab World, I highly recommend Iyad Al-Baghdadi's 20 minute presentation, It Happened Before, It'll Happen Again. It addresses this very question, and I daresay does a better job of it than this answer does.

^{* - It may be noted that these 5 things were a cultural feature of the entire Arab World, not just Tunisia. Hence the "Arab Spring".}