As you probably have an inkling of this already, British India was partitioned on religious lines (the princely states however were just granted independence^1 and the French and Portuguese colonies joined India at a later date). The British colonial authorities decided that provinces with a majority Muslim population would go to an independent Muslim state called Pakistan, while the rest would join a secular India. There are obvious problems with this arrangement as you noted in your question; one of the majority Muslim provinces was East Bengal, which is on the other side of the Indo-Gangetic plain from the rest of the Muslim majority provinces. Another problem is that there were (and still are) significant Muslim minorities across India, especially in the southern and north-central provinces. Were these Muslims supposed to just leave their homes and head to the "appropriate" country? Conversely, were the non-Muslims in the new Pakistani provinces expected to move to India? As it happens, nearly fifteen million people did emigrate and in the ensuing chaos perhaps as many as two million died. Yet another major problem is that the idea of Pakistan is predicated on the belief that all Indian Muslims are the same. As you would expect of a large and diverse region such as India/South Asia, this belief is naive at best and has been demonstrated to be so by subsequent events. The 1971 Bangladeshi War of Independence (in which they were aided by Indian intervention), and the current and sizable support for nationalist parties and insurgencies in Baluchistan and the Pashtun frontier are unsurprising reactions to the political dominance of Punjab and Sindh.

Now, why did the British colonial administration settle on a religious based partition?

It must be stated that throughout Indian history Muslims and other religious groups coexisted relatively well, though not without some trouble. Much of the communal violence and growing distance between the various religious groups occurred under European rule for two main reasons. The native religious groups across Asia and Africa found themselves in a world where they were dominated by Christian European powers; they had to understand just how this had happened and how they could solve this problem. Unsurprisingly they came to a variety of different conclusions and solutions, and new reformist religious movements appeared. The other main reason concerns the practical application of independence itself; many native groups wanted independence but just what exactly would independence look like and to whom would it apply. Besides these two major reasons you also have traditional ‘divide and rule’ tactics on the part of the colonial government which further contributed to the communal divide. Perhaps the most visible ‘divide and rule’ tactic was the establishment of religiously based separate electorates for the Indian elections in 1909, which the newly-founded All-India Muslim League vigorously campaigned for. However, prior to World War Two, the League were arguably an unpopular party among the Muslim electorate; they only won 109 of the 482 Muslim seats in the 1937 provincial elections. They gained most of their seats in provinces where Muslims were a minority, while the majority Muslim provinces voted for regional parties that tended to focus on class issues. Additionally, the idea of a separate independent Muslim state only appeared during the 1930s among the intellectual leadership of the League; it didn’t become official policy until the Lahore Resolution of 1940. The event that most changed the fortunes of the League was the outbreak of World War Two; the British colonial government declared war on Germany without consulting the native branch of the Indian government. After failed negotiations between the Indian National Congress and the colonial administration, Congress resigned their positions from the provincial governments they controlled (which was almost all of them). The League however offered their unconditional support to the British, who began to see the League as not only the sole representatives of Indian Muslims, but as the main representative of the Indian independence movement. With British support, the League campaigned vigorously throughout the war for an independent Pakistan. In the post-war provincial elections of 1946 the League massively increased their share of votes and seats; they became the largest party in Bengal, Sind, and Punjab, forming a government in the former two. By this point, Partition was practically a done deal.

1 After being granted independence the princely states were expected to join the religiously appropriate country. Almost all of them did, yet there were notable exceptions: Junagadh located in Gujarat, with a mostly Hindu population and a Muslim ruler, attempted to join Pakistan but unrest from Hindu subjects and Indian intervention led to Indian annexation; Hyderabad, a very large southern state with a similar situation to Junagadh, though the prince chose to remain independent, leading to Indian annexation; and Jammu and Kashmir, the events surrounding which have famously resulted in numerous wars between Pakistan and India. Kashmir was a reverse of Junagadh and Hyderabad, comprising an overwhelmingly Muslim population ruled by a Hindu.