What does this mean?
Well, let's take it into parts.
The Judicial power of the United States
The Courts of the United States...
[S]hall not be construed to extend to any suit in law or equity, commenced or prosecuted against one of the United States by Citizens of another State[.]
First let's take a look into history books:
Before the 11th Amendment, the Supreme Court went by Section 2 of Article III which allowed Federal Courts to hear the disputes between a State & Citizens of another State or Citizens or Subjects go a foreign State.
SPEEDY TRANSLATION: Basically if an American had a problem with the State next door, or if a Country had an issue with one of the States, all they had to do was bring the problem to the Supreme Court to reach a solution.
Now, the main reason this Amendment even became apart of the Constitution was because of the Chisholm VS. Georgia case.
At that time, citizens of one state could sue another state via the Supreme Court.
Why did Chisholm want to sue the State of Georgia?
Because they borrowed money from him during the Revolutionary War and basically he wanted his money back.
When the Supreme Court called the State of Georgia, they were basically like uh no I'm a State, this entire case is beneath me, this is not happening.
Well, it happened.
A South Carolina man sued the State of Georgia.
Then, many others began filling lawsuits against States - including a British man who sued Massachusetts for taking his property that he owned in America.