How did the Great Rebellion break out among the British-led Indian army?

It is important to note that India, prior to the Great Rebellion, was governed and controlled by the British East India Company rather than Great Britain itself. To consolidate Company control over India, the Company used Indian troops to defend their interests. These soldiers formed the backbone of the East India Company's own military forces, and they were the troops who mutinied, starting the Rebellion.
As has already been addressed, there were already tensions within Company-ruled India, and these tensions were inflamed by rumors of rifle cartridges greased with animal parts. These rumors led to a number of confrontations between sepoys and their officers, which culminated with the mutiny at Meerut. Knowledge of these events inspired similar acts of rebellion, and thus the unrest spread.
The Great Rebellion began with the sepoys and spread toward other disaffected portions of the Indian population, such as peasants and local political elites. Thus, the rebellion's scope grew larger than the soldier class which launched it. In the end, the British government itself intervened to quash the revolt.

A series of actions led to the Great Rebellion. First, the British disrespected Indian rituals and traditions. They encouraged the spread of Christianity and tried to do away with the Hindu and Muslim religions. The British also introduced western education, which focused on European ideas and disregarded the Indian education system.
Another factor was the new Enfield rifle. After the weapon was introduced between 1856 and 1857, rumors spread that the cartridges used to lubricate this gun were made of cow and pig parts. This angered the Indian troops even more.
Following the rumors about the Enfield rifle, some Indian troops in Meerut refused to use those guns and were sentenced to jail as a result. This was the last straw, and the Indians felt it was time to do something about it. On May 10, 1857, the Great Rebellion commenced after Indian troops shot their British counterparts and went on to seize Delhi.
http://www.bbc.co.uk/history/british/victorians/indian_rebellion_01.shtml