Brief Summary
The Set-Up
A large sweep of Chinese immigration was followed by an economic downturn (these two things are entirely unrelated, too). White laborers wanted to protect their jobs from outsiders, and created enough political pressure to make that happen.
The Text
The law starts out by saying Chinese laborers constitute a danger to the United States. Yup; it just comes right out and says it. Then it says they can't immigrate for ten years.
There's only so many ways to say, "Hey, all those Chinese people who want to live and work in the U.S.? No thanks." This law tries all of them. The meat of the law is in Section 1, which is that bars Chinese people from immigrating for ten years.
The rest of it is spent on enforcement, including a system of monitoring Chinese people already in the country. It also points out how ship captains are supposed to deal with Chinese passengers, since this was before commercial air travel.
TL;DR
No more Chinese people can immigrate to the United States. #sorrynotsorry.