The 1971 war was deeply tied to the Bangladesh Liberation War, and is really part of that conflict. Bangladesh was once part of Pakistan and was known as "East Pakistan" while modern-day Pakistan was known as "West Pakistan". The two were obviously separated by India in between and have a number of cultural differences, including language. Political power was concentrated in West Pakistan and most of the greater government officials came from there, leaving East Pakistanis feeling disenfranchised and under-represented. Furthermore, the Pakistani government enacted reforms to ensure that a Bengali party could not win an election and form a government. This inflamed tensions which already existed.

Eventually, unrest led to the army being sent in to East Pakistan and an independence declaration from Mujibur Rehman, who formed the a provisional government in Bangladesh. This was a Pakistani Civil War, but as a result of the fighting and atrocities committed against civilians, many Bengalis had to leave their homes and went to India as refugees. India first simply opened its borders to refugees and declared their support for Bangladesh, but their involvement increased with time. First, they began arming Bengalis to fight against the Pakistani Army as insurgents. Pakistan was obviously not happy about India arming and training terrorists to fight against them, and their was a great deal of public outcry in Pakistan against India. While doing this India was preparing to go to war against Pakistan.

Knowing that war was inevitable, Pakistan launched a preemptive strike against Indian air bases, and India was thus dragged into the war (although they would have entered on their own soon anyway). The Pakistani tactic was ineffective, and after just 13 days, Pakistan surrendered. This resulted in independence for Bangladesh, and created Pakistan as we know it today.

Motives for Indian intervention depend on who you ask, but for my money it was a bit of everything. First, India saw an opportunity to substantially weaken its greatest enemy. They'd already been to war in 1947 and 1965, and are still enemies to this day. In losing Bangladesh, Pakistan was obviously considerably weakened. Second, it was a legitimate humanitarian crisis, and India had the power to intervene and stop it. They could easily come away as the "good guy" through intervention. Third, it was an economic decision. Caring for millions of refugees was expensive, and no country wants to deal with a prolonged refugee crisis. Military intervention meant they would no longer have to care for all of the refugees, who could then return home.