What drives someone in India to come to the US, rather than improving their situation at home?
Different reasons depending on the route one takes. For those like me that took the educational route, perhaps it was disillusionment with things back home coupled with having a dream and a getting new opportunity to start things fresh and try something that wasn't possible back home. As India has developed, and as there is awareness of the "cons" of US life too, those like me are in the minority.
What challenges did they have getting here?
As an evil fox character in my daughter's favorite TV show would say, "Oh man!". I'd sum up the challenges for legal immigration as a potent mix of unreasonable visa quotas and dealing with a slow, needlessly complex system (that almost makes illegal immigration look good by comparison). As an example of the former, in some cases people have graduated from US schools with a job offer which they couldn't accept because the visas ran out on the very first day and the resulting "lottery" didn't favor their application. Why would today's youth choose to suffer this when other developed countries (eg. Canada) will give you a green card out of college even without a job offer...
As an example of the grinding wheels of the system, I'm a PhD scientist doing very specialized (not run of the mill IT) work, and my application took 6 years to clear, including many frustrating delays and redundant requests for fingerprints or documentary evidence already received by them. It seems USCIS is accountable to no one in this matter (especially not their legal immigrant customers who keep shelling out hundreds of dollars for employment and travel privileges while waiting for cards to arrive year after year). Again, why would... as above.
How have they secured their stay?
With a green card and a career, I suppose... but I'll contort the question to answer a related one: why don't I go back, now that things are better. I believe we might some day, but for now what we gain by being here still outweighs (somewhat) the high cost structure for homeowners/parents, losing out on being near parents, and other reasons for being in India. For many many of my friends who've been trained here but have different constraints/expectations than ours, the answer has been clear cut, and they have indeed returned to India.
The best way to accelerate a Macintoy is at 9.8 meters per second per second.