What is the importance of the Indian Institutes of Technology
Five campuses of the Indian Institute of Technology (Kharagpur, Bombay, Madras Kanpur, and Delhi) were established in the late 50s and the early 60s with the intention of producing engineers and leaders who would lead India's industrialization and modernization. These institutes had significant funding from foreign governments (USSR, USA, Germany) and organizations (UNESCO). University of Roorkee was later converted into an IIT and IIT Guwahati was established in 1994.
In a country where most places still don't get 24-hour electricity, IITs quickly became the mecca of education. Thanks to a healthy amount of government funding, they had excellent residential facilities, recreational facilities and lab facilities. They attracted the best faculty, and developed good industry contacts which led to some of the best job opportunities.
For some inexplicable reason, Indians typically value engineering and medicine above most other professions. So most bright students are encouraged to take up either of these two. For perhaps this reason, the amount of attention and care given to the IITs was not extended to liberal arts and pure science colleges. IITs soon became the place to go to, to get a quality undergraduate education.
Their legendary status was further established by a draconian entrance examination, the IIT Joint Entrance Examination (IIT JEE) which at its peak, had a selection rate of about 0.1%. This has recently doubled as the government has established 8 new IITs (Bhubhaneshwar, Gandhinagar, Hyderabad, Indore, Jodhpur, Mandi, Patna and Ropar). Institute of Technology Banaras Hindu University (IT-BHU) has also been accorded IIT status. About 400000 people take the entrance exam every year, and about 8000 get through. You need a rank in the top 3000 to get into IIT Bombay, and Computer Science at IIT Bombay closes below rank 100. So if you want to study computer science at IIT Bombay, you basically need to be in the top 0.025%. Understandably, the competition to get into the IITs is cut-throat
So, given the level of hype around the IITs, a whole industry has cropped up - IIT JEE coaching classes. There are entire towns (Kota in Rajasthan) which just have a huge number of coaching classes and almost nothing else. These coaching classes further propagate the "legend of the IITs" which has led to the cult status that the IITs currently enjoy in Indian society.
It should be noted that these numbers (0.1% etc.) aren't as impressive when you take into account the fact that almost every aspiring engineer in India takes the exam, while not every student in the US would apply to an MIT or a Stanford. And there are other institutes of comparable standards - BITS Pilani being one of them and maybe one or two NITs.
Of course, that is not to say that IITs are all about the hype. You need to be very smart, very hardworking or very lucky to get into an IIT (At the risk of offending some IITians, I daresay that luck has become much more of a factor since the exam became multiple choice objective). Because the IITs get the best students in the country, the best faculty are attracted to it, the best companies come to hire here, and foreign universities also recognize the name (thanks to 50 years of the steady diaspora of IITians, which incidentally, is another thing detractors of IITs take issue with).
All these people - engineers, managers, academicians, who are alumni of IITB have ostensibly done well in their chosen fields, which is the reason people still want to hire IITians. And as a result of all this, IIT has become the biggest brand name in Indian education today. Which is why you hear about them all the time.