How did such a relatively minor event as the assassination of an archduke start World War I?

The simplest answer might be something like "the assassination itself caused only the Austro-Hungarian declaration of war against Serbia, not the World War itself which was the result of a chain reaction and long underlying factors". To illustrate this, I could add something based on a very relevant comment by user Michael.

"According to Erenburg, when France was mobilized in 1914 some small store owners put out signs on their doors: "out for the annual mobilization." Major European countries were itching for a war, it just needed a pretext. – Michael Dec 21 '16 at 18:57"

This seems to be backed up by an article I found on the internet a few years ago, which said that in France old mobilization posters were found on which the date 1914 was discovered to have been originally "1904" but changed into "1914". This would suggest France had been preparing for war long before it actually happened (I may have saved the article at the time or I may have not - I have no idea and it would be difficult for me to find out at the moment, but a search might resolve this).

The reason was the system of alliances.

The conflict started as a conflict between Austria Hungary (who wanted revenge for the killing) and Serbia.

But Serbia was allied with Russia, Russia was allied with France, and France was allied with Britain. Also, Austria-Hungary was allied with Germany.

So when Austria-Hungary threatened Serbia, Russia declared war on Austria-Hungary. Germany tried to "warn off" Russia, but ended up at war instead. France declared war on Germany, using the occasion to get revenge for 1871.

In theory, Germany could have stood on the defensive against France and joined Austria-Hungary against Russia, but another poster pointed out the dangers of such a poster, a belief that was widely held (on both sides) at the time.

So when Germany took the offensive by attacking France via Belgium, Britain went to war against Germany to protect Belgium.

The war was already built-in, because Germany was faced with a strategic conundrum/objective circumstances:

• If Russians enter the war, Germany risks losing it, fighting France and Russia on two fronts

• Russians (if I recall the contemporary estimates correctly) would take ~2 weeks to call up its forces and actually be able to attack.

• As such, Germany's only (seemingly, at least) viable option was to attack France immediately, and force it to surrender, very quickly, before Russia had a chance to offer a second Eastern front.

Therefore, the moment Russia acted aggressively past Arch-Duke's assassination and Austrian saber-rattling towards Serbia, Germany basically had only one path forward: attack France ASAP and hope to implement the Schlieffen Plan and take France out of the war before Russia got there.

As German's government saw it, the only other alternative was to wait for Russia to activate its army, attack Austria, and then together with France attack Germany.

Whether the assassination of an Arch-Duke is a 'minor event' is a matter of opinion. In this particular case it was also the assassination of the Heir to the Austro-Hungarian throne, the Inspector of its armed forces (implying probable role as Commander in Chief in the event of war) and a close relative of the Emperor. Arch-Duke Franz-Ferdinand was all of those things. So the murder of the Arch Duke was a big deal from Austria-Hungary's point of view. Austria-Hungary was still considered a major power in those days, with a proud history.

However, Austria-Hungary's position as a great power, and even its continued existence, was threatened. It had lost wars to France and Italy in the 1850s and Prussia in the 1860s, and as a multi-ethnic empire in an age of nationalism it was threatened by possible break-away movements among its many different nationalities. It could not therefore be seen to be weak in the face of terrorist assassins from what were generally considered more minor and backward peoples like the Serbs and Croats.

That may help explain why the assassination led to war between Austria-Hungary and Serbia. Why did it lead to a general European, and eventually World War?

Well, people write books about the causes of the First World War so this is very far from a complete answer.

However, it had been quite common in the Seventeenth and Eighteenth centuries, and Napoleonic Wars, that once a War began somewhere in Europe other powers joined in if they saw an advantage/ a need to stop the largest power getting any more powerful from their own security. E.g. the Wars of the Spanish and Austrian successions in the Eighteenth Century and French Revolutionary and Napoleonic Wars 1792 - 1815, when other powers often co-operated to stop France becoming too dominant, or used the fact that others were occupied fighting each other to grab territory.

By 1914 the power that threatened to dominate the rest was Germany, so the choice for the others was whether to combine to resist German dominance (as Britain, France and Russia did) or to fall in with it as a junior partner on what they hoped would be the winning side (as Austria-Hungary and Turkey did).

A significant issue that I believe is linked to this is why, against precedent, before 1914 many of the Great Powers had formed long-term treaty alliances even in peacetime (as France had with Russia, Austria-Hungary had with Germany and Britain with Japan) by which they were committed to support each other in the event of a major war. This made it more likely that a war between a couple of countries would spread.

Added to which:

German war plans hinged taking advantage of their central position and good railway and road networks to assemble their army and strike before the other powers were ready, so they would not wait around for long discussing possible diplomatic compromises.

Britain (then under a relatively pacific Liberal government) was genuinely ambivalent about whether it would or should stand by France in the event of war; consequently there was not a clear enough warning to Germany that Britain would fight on the French side if Germany started a war.