Note: I am drawing my information from Sleepwalkers: How Europe Went To War In 1914 by Christopher Clark.

This is a question that requires a long answer, and I'm quite certain I will miss something - I invite anyone who can to fill in the blanks.

The political situation in Serbia at the time of the assassination of Franz Ferdinand was complicated. The previous royal family of Serbia had been assassinated and replaced by a new dynasty, the Karadjordevic's, and the regicide faction was still present in Serbian internal politics, often pitting themselves against more moderate politicians. Beyond this, you had a growing nationalist movement, represented both by the Narodna Odbrana (formed in response of the Austrian annexation of Bosnia-Herzegovina) and the Black Hand, which were involved with one another, and who were infiltrating the ranks of the governing body of Serbia, turning the tide of both their internal and foreign policy towards a more nationalist and belligerent bent.

Add to this that Serbian politics were being meddled in by the Russians, who were hoping to strengthen their position against Austria-Hungary AND at the same time hopefully grabbing some territory from the crumbling Ottoman Empire (they desperately wanted access to the Mediterranean for their Black Sea fleet), and you've got the recipe for a very volatile situation.

There was a Pan-Slavic movement - with the hopes of forming a state for the South Slavs, whether they were living in Bosnia, Serbia or within the Austro-Hungarian empire - and it seems the Serbian government under Nikola Pašić did little to stem the tide. Narodna Odbrana and the Black Hand were pushing for a more belligerent stance against Austria-Hungary, both through influencing Serbian politics and through acts of terrorism, such as the one committed by Gavrilo Princip and his accomplices.

While I sincerely doubt that Nikola Pašić himself hatched a plot to assassinate the Archduke - though the Serbian authorities seemed aware of the plot far enough in advance to issue vaguely worded warnings to their Austrian counterparts - the assassins were armed with weapons drawn from the Serbian army's main armoury (which I can't for the life of me recall the name of at the moment), and after the assassination, the Serbians did much to obstruct and hinder the Austrian inquiry into the matter, possibly out of fear of the Austrians finding any connections between Serbian politicians and the assassins.

Clark seems to be of the opinion that Serbia was not 100% directly responsible for the assassination plot, but that a lot of the blame for it can be laid at their feet, and that they did little to ensure there WASN'T an assassination.

How responsible? Well, the country existed, which was very much an annoyance to Austro-Hungarian interests in the Balkan because it represented a free Slavic country that the nationalistic Slavic population of the Habsburg lands could gravitate towards. And in the decade before the start of the war it had become especially annoying to Austria-Hungary, who saw itself as a major Balkan power.

In 1903 a conspiracy from within the Serbian military had assassinated King Alexander I Obrenović for reasons which included being too friendly with Austria and brought Petar I Karađorđević on, who was a proponent of democracy and liberality and pretty much against Austrian Balkan interests, much to the joy of most of Serbia. The leader of that conspiracy was a man named Dragutin Dimitrijević, at that time colonel in the Serbian Army, who now quickly became a national hero for this deed.

Then 1912/13 we have the two Balkan Wars, in which Serbia expanded greatly. Austria-Hungary didn't want a regular free Serbia on their Balkan, and they certainly didn't want a strong one. Already after the Serbian success of the First Balkan War in 1912 there were serious talks of attacking Serbia and thus launching a World War but the Germans concluded their military wasn't ready yet.
In Serbia, national hero Dragutin Dimitrijević's planing involvement in the Serbian Army's success netted him the post of Chief of Military Intelligence after the war.

Already in 1911 Dimitrijević and nine other men had come together to form a secret society with the goal of uniting the Slavs under the flag of Serbia, by means of violence where necessary. Essentially a terrorist group, it was recruited in large parts from the Army and organised in small, independent cells that only had a single contact to a higher tier until eventually everything came together with the Executive Committee in Belgrade led by Apis, official code name of Dimitrijević. This group called itself Ujedinjenje ili Smrt (Unification or Death) or Black Hand unofficially.

So while the Black Hand was never officially supported by the Serbian government it was, through its members, in some very influential positions in it, most importantly surely Apis, who was after all the main hub for all secret intelligence involving Serbia. It's also worth noting that the acting conspirators in the 1914 assassination were not members of the Black Hand but a group called Mlada Bosna (Young Bosnia) with the goal to free Bosnia-Herzegovina from Habsburg rule and unite with Serbia. There were for obvious reasons shared interests and members between the two groups but I think it's nevertheless important to mention the disctinction since Princip actually failed to get into the Black Hand at one point. And while the Sarajevo attack was by all intents and purposes run by the Black Hand (training and supplies) it was actually the idea of Princip's circle of nationalists.

The daily thing you read yesterday was reference to a secret cable from the investigators in Sarajevo to the Austrian Foreign Minister Count Leopold von Berchtold:

Hardly any room for doubt that Princip, Grabez, Cabrinovic smuggled across border with help from Serb customs… However, no evidence of complicity of Serb government ministers in directly ordering assassination or in supplying weapons.

As for why Serbia complied with Austrian demands... They just didn't want war. They had just come out of the Balkan Wars victorious and needed to sort their gains there out and with the rising Slavic nationalism in Austro-Hungarian territories it was not unreasonable to try and let that play out by itself since the dual monarchy was already considered unstable in most European circles.

Sources:
Newton, Michael - Age of Assassins: A History of Conspiracy and Political Violence, 1865-1981
Owings, Dolph - The Sarajevo Trial
Lazlo Csorba, John C. Swanson, Peter Hanak, et al. - Österreich-Ungarn, Das Habsburgerreich von 1867 bis 1918