I wrote a little bit about the "riot" here, though I did not address the "why". To summarize, the audience included both posh, older, conservative upper-class people, and a younger, "bohemian", avant-garde crowd. The former got personally insulted by Diaghilev's and Nijinsky's disregard for ballet conventions and by StravinskyDiaghilev's "cacophonic" music. They started booing and cat-calling, an old tradition in French theatre, though by the turn of the century demonstrations of audience's displeasure were considerably more civil than they used to be (also: Tomatoes). The younger crowd responded in kind, which resulted in a brief (but stressful) "fight" that did not prevent the performance from ending peacefully, and later performances went well too. Moore (2009):

Nijinsky had worked with his dancers to create a body shape for the entire ballet that seemed to be the exact opposite of the openness and lightness of classical ballet. Toes and knees were turned in rather than out in the traditional ballet pose, feet were mostly flat on the floor not on point, and the energy of the dancers’ bodies was directed downwards into the ground rather than up into the air.

Just as the curtain rose, the music underwent a sudden and shocking transformation: the orchestra took up the plucked violins’ rhythmic pulsing from a moment before and amplified it into a loud, stabbing, dissonant dance, with unpredictable slashing accents. The circle of men, crammed closely together, dressed in heavy tunics and fur hats, began a stamping, jerking dance in their knock-kneed position, each one mimicking the rhythms of the orchestra but each having his own individual variation like some kind of demented machine. Another rule of ballet was broken in that, when the curtain opened, the performers were not facing out to the audience in the generous, open display of classical ballet, but their bodies were turned inwards and downwards in the tight, huddled circle. It was at this point that, in Stravinsky’s words, the storm broke. Those who were appalled by the ballet no longer felt restrained, and those who were defending it were equally energetic. The company’s ballet master Sergey Grigoriev described what happened next: ‘A section of the audience began shouting its indignation; on which the rest retaliated with loud appeals for order. The hubbub soon became deafening.

Writer Jean Cocteau, a friend of Stravinsky, Nijinsky and Diaghilev, who was there and had to comfort the three men afterwards (Cocteau, 1918, Le Coq et l'Arlequin):

The audience played the role it was supposed to play; it immediately revolted. People laughed, booed, hissed, imitated the cries of animals, and perhaps they might have got tired in the long run if the crowd of aesthetes and a few musicians, carried away by their excessive zeal, had not insulted and even jostled the people in the boxes. The uproar degenerated into a free for all. Standing in her box, her tiara askew, the old countess of P. brandished her fan, and shouted, her face red: "This is the first time in sixty years that anyone has dared to make fun of me." The good lady was sincere; she believed in a mystification.

It is clear that part of the audience and critics reacted strongly and negatively to the piece, but there was no repeat of the premiere's scuffle and the Sacre was quickly considered to be a classic.

Edit: fixed stupid mistake