Just to clarify- do you mean Indians or the indigenous peoples of the Americas? Those are each very different stories.

If you mean Indians in India, they weren't weak, just disunited. The British arrived at a lucky moment in Indian history, when the Mughal Empire of Aurangzeb was drained after a 40+ year war with the Maratha confederacy and a 26-year war with south India.

The Maratha war was harsh, with no winners. Territory kept changing hands as one side captured a fort only to lose it a few years later, then recapture it again. It drained the empire of money and soldiers and set up bitter rivalries that the British would later use to their advantage.

In the last 26 years of his life, Aurangzeb waged war on south India. Although he eventually won, it pretty much destroyed the Mughal Empire. Millions of people died, the Deccan was stripped bare of farms, all food surpluses of the past years were used up. Famines and plague followed.

Meanwhile, this was also the period of the Sikh rebellion under Guru Tegh Bahadur and Guru Gobind Singh, and the Pashtun rebellion.

When Aurangzeb finally died, his empire was a wreck. There was no money left to fight the various rebellions that broke out. The Marathas immediately launched attacks and chewed off large portions of the empire. Dozens of nobles and petty rulers who had previously been part of the empire broke off, declared independence, and began fighting each other to gain more territory. A dozen different short-lived rulers followed Aurangzeb, but within a few decades of his death, his empire which had covered almost all of India and Pakistan was reduced to just the city walls of Delhi.

Bengal, one of the provinces of the Mughal Empire, had been loyal to the Mughals until the death of Alivardi Khan, the Nawab of Bengal. His son and successor, Siraj-ud-Daula, tried to continue the tradition, but his own courtiers no longer recognized his right to rule, which was awarded by the Mughal Emperor. The Mughal Emperor had no power left to enforce his will in Bengal. Court intrigues forced Siraj-ud-Daula into war with the British, who had been awarded trading contracts and some territory in Bengal by Aurangzeb. His own uncle Mir Jafar betrayed him to the British, and Siraj-ud-Daula lost the battle of Plassey to the British.

Gaining Bengal was the start of British rule in India, and from there, the Brits expanded, setting up one little kingdom against another, sharing the spoils, then coming back to take the rest from their former allies, moving on to the next. By this time, the Mughal Empire had collapsed, India was full of dozens of small kingdoms with long standing hatreds that had been simmering under Mughal rule. They hated each other more than they hated the Brits, who had no prior history in India. So the Brits eventually came to occupy most of the subcontinent.