From what I have read in my history books the picture that was painted seemed to me that for the people that became workers the living standards declined, but I wanted to get a better understanding of this since the explanations were not detailed.
So how did they compare? Naturally the comparison can only be made with the workers and farmers of the same country (Britain, USA, Germany/Prussia, France, etc.).
Well, you are on to something when you say you have to compare workers in the same country, as "farm" and "industry" cover a lot. If you pick New England or New York's Hudson Valley, circa 1840, a farmer could be living rather well- that is to say, he and his family could be well-fed, and have a market of a larger city near by to sell what they could grow and raise. He and the family were working very long hours: indeed, the motivation for mandatory public schooling was that, if allowed, most farmers would keep their children at work on their farm. But when you look at the working conditions, working hours of the Lowell MA mills, circa 1840, they're quite long as well, and the wages were rather meager. And factory conditions generally could be grim enough: one of the tropes of the defenders of slavery in the South was that slave-owners treated their slaves better than New England mill owners treated their workers ( I don't think anyone's ever found this to be true) and even the optimistic books of Horatio Alger never had a hero succeed by starting off working in a factory.
So, why leave the farm? The key difference, here, is that farming needs land and there's only so much of it, and once you have enough people to work a farm you can't expand it without getting more land. So, there's no good way to employ the excess population. The Lowell mills were able to attract a workforce of young farm girls who had few prospects: they were not going to inherit their father's farm, and they might not feel like trying to marry the owner of another. And there would be sons, too, who were not in the position of getting the farm or perhaps only able to inherit a part of one, and might be faced with a choice of working on a very poor piece of land, working for wages on another farm that they'd never inherit, or trying a job in a mill.
Another thing that the urban mills offered the workers that farm life did not was a social life. By their nature, farms are out in the county and before the advent of pickup trucks farmers and farm workers spent much of their time isolated. The Lowell girls had a library and reading rooms, and other girls to talk to. They were able to put out a periodical of their writings, the Lowell Offering. In this way, you could say that the factory standard of living could be much higher than the farm's.