I understand that the British government took over all Indian territory and trade held by the East India Company. Why did they choose to this task onto themselves instead of outsource it to the EIC?
The immediate cause of the creation of British India from the former EIC territories was the Indian rebellion of 1857 . This was a widespread, violent rebellion against company rule, which was eventually put down. After the dust settled, many changes were put in place:
The rebellion remains a very emotive subject among certain circles, so I won't go into too much detail about it, but one interesting indication is that in India, it is known as the "First War of Independence" while in the UK it is usually called the "Indian Mutiny". You can read more by doing a quick Google search.