I hear this phrase quite often, but I've never heard the origin of it. It would be also nice to find out what it means, exactly.
The New Oxford American has this:
rest one's case conclude one's presentation of evidence and arguments in a lawsuit.
• humorous said to show that one believes one has presented sufficient evidence for one's views.
It predates Perry Mason by a bit. My New Shorter Oxford English dates the following definition of rest from the mid-1800s:
c. Voluntarily conclude presenting evidence in a law case. US.
It was likely popularized by old-time legal TV shows, such as Perry Mason (I don't recall if Perry ever said that phrase, but it was common in that type of show.)
During a court trial, the prosecution presents its case, then the defense defends against the allegations made. At the end of each part, that side announces that they are finished by saying that they "rest", as in "the prosecution rests", meaning "we on the prosecution side have nothing more to say; we have made our case."
Old-time court drama TV was full of dramatics and grandstanding, so some TV personality or another probably made-famous the phrase by presenting undeniable proof and then announcing to the courtroom "I rest my case!"
The implication is: you should need no more evidence; everything is explained by what you just heard.