I was about to head to sleep, and then I had to answer this one.

To be more specific, it was a group which was later identified as "Young Bosnia" which assassinated him. The Black Hand assisted the Young Bosnians. It's a common misconception, according to sources I've read, to attribute the assassination to the Black Hand. Young Bosnians were an offshoot of The Black Hand at best.

Moving along! The main reason for the assassination has to do (as most conflicts do) with international tensions and radical ideology.

Bosnia & Herzegovina (B&H) had been occupied by Austria-Hungary at the Congress of Berlin in 1878. Prior, it had been ruled by the Ottoman Empire. Now, there was a pretty weird uncertainty in all this. The area was occupied, but not annexed (at first), and the people there considered multiple routes. They could attempt to rejoin the Ottoman Empire, join Austria-Hungary, join a federal Yugoslavia, or join a new Greater Serbia.

In 1908, that choice was made for them. The Austro-Hungarian authorities fully annexed the region, partially to prevent Ottoman control being returned, and partially to ward off a possible Serbian invasion (these are likely reasons, I don't pretend to know exactly everything!).

Serbians responded with counter-literature, through the Narodna Odbrana (National Defense) organization. They, with over 5,000 members over a few months, were augmented by other groups that wanted to create a Greater Serbia. They spread counter-hegemonic literature, essentially saying that joining the Austro-Hungarians was bad.

One of the main groups augmenting National Defense was the Black Hand, an offshoot of the Serbian army. But they weren't the end of it. The increased radicalization of people, including young men who took on a more revolutionary attitude, led to the creation of Young Bosnia, the movement which Gavrilo Princip belonged to. They essentially focused on "Slavization over Germanization, fighting against attitudes of servility, and removal of alien influences". They created a sense of national identity and fervor for separation from Austria-Hungary that allowed them to effectively mobilize members in support of their cause. They were attempting assassinations before Princip, too.

In June of 1910, a member attempted to assassinate the new governor of the Sarajevo area, General Varesanin. This failed, and Zerajic (the assassin) missed 5 times before killing himself. His body was found. This "martyr" status and sense of sacrifice only invigorated the Young Bosnians. It's important to note that not only Bosnians or Serbs were feeling these radicalized ideas. Croatia faced demonstrations at a university in 1912, especially against the rule of Count Cuvaj. The Young Bosnians tried (and failed) to assassinate him as well. The person responsible for the attempt did manage to kill a policeman, and the Chief of the Croatian Department of Information before being caught, however.

The Austro-Hungarians, in light of this, were quick to crack down on student protests. Soon, there became an even larger disconnect between the youth and those in power, which made it even more dangerous. Trouble was brewing, by this point.

Archduke Ferdinand decided to visit Sarajevo in 1914 to demonstrate the Austro-Hungarian power/interest in the area, to inspect the area (as was his job) at the governor's request. It didn't seem like a big deal to anyone involved, just a formality and indication of care. Now, Gavrilo Princip's cell of the Young Bosnians, so to speak, was led by Danilo Ilic. Princip came up with the idea, upon hearing of the impending visit by the Archduke, to attempt to kill him. Princip ran it by a friend (Nedeljko Cabrinovic), and then recruited another: Trifko Grabez (a schoolmate as well). Now, keep in mind, Princip was still in high school at this time. Young Bosnians is really Young Bosnians. Princip approached an old friend/soldier, Ciganovic, but here the story gets vague and murky. We rely on 14-years-later accounts from a co-conspirator, basically.

It seems Princip and co. stumbled upon a society of sorts called Death or Life, which Ciganovic and friends founded. They had plenty of enmity towards the Austro-Hungarians, and even met on the site of an assassination in 1868 of a Serbian prince (supposedly, this happened thanks to the Austrians helping the assassins). There, it's recalled that Princip and the others were proposed as the future assassins, and asked to join the society.

They were trained, given guns and weapons, and in the meantime, Ilic (the leader of the "cell" of Young Bosnians) had recruited more people for Princip to get help.

The rest, as they say, is history. The assassination attempt (though initially failed), ended up working by mere luck on the part of Princip, who saw the Archduke's car after the bomb attempt failed (the bomb was thrown and missed the car) and fired multiple bullets (two or more) at the Archduke and his wife. Both were killed. The intent was to make a statement; martyr oneself for the cause of a greater Serbia and freedom from the Austro-Hungarians, who were viewed as oppressors and aggressors unwanted in Bosnia. The message was sent, without a doubt, but it's doubtful the Black Hand or Young Bosnians knew exactly how it would end up, regardless. At best, they likely hoped for revolts, rebellion, and the ceding of Bosnia to the Serbians. The idea that this would lead to the conflagration it did is definitely missing from their writings, unsurprisingly!

Jackson, P. (2006). ‘Union or Death!’: Gavrilo Princip, Young Bosnia and the Role of ‘Sacred Time’ in the Dynamics of Nationalist Terrorism. Totalitarian Movements & Political Religions

Edit: Just gonna take a moment to express my astonishment to waking up to 13 responses. I've never talked to many people on Reddit, this is kinda weird. Now, to the answers!

The whole thing may not have happened if it weren’t for a sandwich. Before I go further on this, I should state that I am, by the preceding sentence, at risk of contradicting myself. As the reader will find, I am not a fan of what I call the “what-ifs of history”- the suppositional posture of predicting how events in the past may have unfolded but for a change in circumstance. These thoughts can be debated until breath is exhausted, but it is, I believe, a wasted effort. History is what was, not what it could have been. That being said, I cannot help but point out the remarkable coincidence that set the events of a continental war in Europe in motion during the summer of 1914.
Gavrillo Princip, a young man from Bosnia of Serbian nationality was supposedly a member of a group called “The Black Hand.” This was a secret organisation, backed by officers the Serbian military whose aim was to gain independence for Serbians in Bosnia who fell under the rule of the Hapsburg House which headed the Austro-Hungarian Empire. Princip had been tasked, and it is believed that there were others with the same instruction, to assassinate Arch-Duke Franz Ferdinand, the heir to the Hapsburg throne while he was on tour of the Bosnian capital of Sarajevo, with his wife, the Grand-Duchess. Princip, armed with a pistol allegedly acquired from Black Hand sources in the Serbian Army, had waited along the designated route of the royal motorcade, but adoring crowds, mechanical problems with the vehicles in the motorcade and genuine fear for the Arch-Duke’s safety had delayed its progress. Believing he had time to spare, Princip ducked down a side street to buy a sandwich from a local shop. Incredulously, at the same time, the royal procession had taken a wrong turning that placed it not where the potential assassin should have been, but where he was due to hunger and boredom. Shortly after emerging from the shop, Princip was confronted by his intended target. Drawing his pistol, he fired into the open-topped car conducting the royal couple, fatally wounding them both. It is widely understood that the assassination of members of the Austro-Hungarian nobility was the prime catalyst which sent an already politically tense Europe into the most destructive conflict the world had yet seen. What is generally left unexamined is the motive behind the killings and its expected results. A small, secretive group such as the Black Hand certainly was not influential enough to inspire the national leadership of Serbia to pursue the aim of annexing territory from Bosnia being that the latter was under the protection of the more powerful Hapsburgs. The only way the Black Hand felt that its political goals could be met, which was the ceding of foreign territory populated by ethnic Serbians was to act in such a way as to provoke war between Austro-Hungary and Serbia.
The outrage at the death of Franz Ferdinand would be sufficient provocation; and the war aims of Serbia in such a conflict would be this repatriation of traditionally Serbian land and people. The intentions of the Black Hand were certainly not to bring the major European powers of Germany, France and Russia into a general conflict. They perhaps relied on the notion that such a war wasn’t possible based upon an equitable match of military strengths among potential belligerents which had acted as a deterrent against a large scale conflict in the past. Serbia’s link to Russia (and thus by extension, France) and Austro-Hungary’s to Germany would be enough to have such a deterrent for interference from either camp, which would limit the war in scope and participation to the Balkans (the collective geographic term for a number of ethnically and nationally diverse countries which included Serbia and Bosnia).
The error in judgement, that general war wasn’t possible, was that the very same militarism which had been a deterrent factor thus far had become too difficult to maintain in that role. With each of the major powers, establishing strong armed forces had led to the elevation of military leaders to prominent and influential roles within their governments. It was these men who saw what could be gained territorially through open war, and these envisioned rewards of victory were too tempting to pass up. Thus the relatively small political upset between two contentious neighbours in southern Europe was excuse enough for the military influentials of the major powers to convince their respective governments to go to war.

The ensuing conflict, known at the time as the Great War, and to us today due to events which occurred afterwards as the First World War, or World War One, would eventually involve some one hundred and thirty countries, be fought on three continents and most of the world’s oceans and be responsible for the deaths of an estimated fifteen million people. (Source: Introduction, pp 16-19, e-book "If Ye Break Faith" Christopher J Harvie)