Professional drivers can be more experienced at maneuvering through this traffic, while the average person does not feel comfortable with that.
The last reason may be the most complicated. It also has to do with the enormous number of people in India. All of these people need to have jobs so that they can work and support their families. In the United States, someone might feel that there are better options for jobs than being a housemaid or driver, but in India, there are not so many other jobs, particularly for people with less education.
The need is being met by jobs that don't require a college education or skilled training. As more people in India go to college, there are fewer people who want to be housemaids and drivers, so this social and economic system is changing here, too. For now, though, many upper and middle-class people pay to have house help at home.