Can you please elaborate on what specifically you're referring to as sweatshop labour? If you're talking about factory work like that in Shenzhen (Foxconn et al), I'm not sure I'd call that "sweatshop conditions", but I can at least answer your question if that's the context. So, assuming that's what you're referring to:

Let's start by looking at Shenzhen, which is a Special Economic Zone set up under the Reform and Opening (改革開放) period. Shenzhen was established as such at the end of the 1970s, and developed so rapidly from that time that it gained the nickname "the Overnight City". There are other factors such as geography which have contributed to its success, but the biggest factor is that the Central Government took a different free-market based approach. Shenzhen is home to many of the Taiwanese and Chinese factories that make your stuff. If you have an iPhone or a Galaxy, it probably came out of Shenzhen.

Shenzhen also has the reputation of the Overnight City for how quickly the population skyrocketed. Suddenly people from places like Sichuan and Anhui moved in to the city to find work, having fewer opportunities back home, and seeing the promises of economic success in the SEZs.

What were migrant workers doing in China before these factories?

For starters, there were far fewer migrant workers in the past, and certainly before the 1980s, you didn't have people flocking to factories in other cities like you see in Shenzhen. Moving to other cities to find work was difficult and for many it would be impossible. But beyond that, the majority of migrant workers still don't end up in these Shenzhen-style factories. They go to places like Beijing to help on construction projects for the Olympics, or to Shanghai to help with the World's Fair, or to anywhere really to build roads and work in other industries. The Chinese New Year is the worlds largest human migration, and it happens every year, and it covers all of China. Here's a map of the migration. They're doing jobs all over the country in all areas of nonspecialised labour. Every major city has construction going on, and on every site, you'll hear dialects from all over the country. This is much of what the migrant workers are doing outside these places.

What factors lead to them accepting these conditions?

In the news you sometimes hear about issues at Foxconn, the Taiwanese multinational most known for making Apple devices. You hear about terrible work conditions and suicides, which may be what prompted you to think of sweatshops. It's well known that workers in these factories typically have 6-day work weeks (itself not too uncommon in Asia) and long hours each day (again, not terribly uncommon, even for white collar workers).

The first thing to note is that for most of these workers, the worst entry-level job that you hear about most often in the news are only held by most workers for one or two years. After that, many try to find other jobs in the company, in other companies, or they move to another city and do something else entirely. The worst jobs are seen as just paying your dues to get your foot in the door. After that you have more opportunities to find work elsewhere. Many workers just come for one or two years anyway. In the pictures that come out of the factories, you'll notice many of the workers are in their late teens for these reasons. The populations in these places is constantly shifting, and no one stays in one place for very long.

There's a common phrase in Chinese: "For those that cannot endure suffering, suffering will last a lifetime; For those who are able to endure it, it is just a passing phase." The ability to endure suffering is highly regarded. If you know that your job of shining iPhone screens is just for a year and then you'll seek out a better situation, it's easier to put up with that year.

If you are interested in the topic of migrant workers and their conditions in South China, you should read Factory Girls by Leslie Chang. It follows the stories of a number of such workers, giving their personal accounts.

There are also a number of fantastic documentaries that follow the lives of migrant workers.

This post is a little rushed because I'm about just about on the way out the door. If anything is unclear, let me know and I can clear it up.