The full text of the ultimatum can be found here but the real deal-breaker was clauses 5 and 6:

(5) To accept the collaboration in Serbia of representatives of the Austro-Hungarian Government for the suppression of the subversive movement directed against the territorial integrity of the Monarchy;

(6) To take judicial proceedings against accessories to the plot of the 28th of June who are on Serbian territory; delegates of the Austro-Hungarian Government will take part in the investigation relating thereto;

These, particularly the second, were judged by the Entente powers to be an unacceptable infringement on Serbian sovereignty. They opened the possibility of Austrian officials directing a Serbian investigation into a plot that was not (yet) linked to the Serbian state.

In truth, I have some sympathy for Austria given that Serbia had displayed little willingness to cooperate with any investigation into the assassinations. Indeed, it's hard to see how the Belgrade government could have fully supported such an investigation given the degree to which the responsible Black Hand organisation was embedded in the Serbian state and military. For example, after the assassination the Austrians requested that Serbia arrest one of the conspirators (Milan Ciganovic) but he had already been smuggled out of Belgrade by the head of police, who promptly denied that Ciganovic had ever been there. That said, I would have a lot more sympathy for Austria if the entire purpose of the ultimatum hadn't been to start a Balkan war to cut Serbia down to size.

As an aside, Christopher Clark's The Sleepwalkers (which covers all this in detail) notes how relative these things are. The ultimatum of 1914 was not nearly as harsh as the 1999 Rambouillet ultimatum that was issued to Serbia by NATO.

As a followup question, I have read (but can't remember where), that the ultimatum was intentionally made unacceptable, and the Austro-Hungarian Empire simply wanted a Casus belli. Is this true?