The author of the passage is telling you where the quotation comes from, using a formal case citation. Lawyers usually don't use footnotes, endnotes, or parentheses to set off these citations.
Case citations are a sort of shorthand developed by lawyers and judges over the past few centuries. The goal is generally to identify the name of the case, where to locate it, when it was decided, and what court decided it. Although there are a few different styles for formal citations to cases, you don't need to learn much to cover the basics for common types of cases.
How to Read It
Brown v. Bd. of Educ.
Case names usually have a "v." in the middle. Most of the time, the plaintiff goes on the left and the defendant goes on the right. But United States Supreme Court cases are different. Almost always, the Supreme Court reviews decisions of lower courts. They do so in an unusual way: the loser in the lower court asks the Supreme Court to change the outcome. Because the loser, more formally called the "appellant," is the one initiating proceedings in the Supreme Court, they go on the left. (To learn more, look into "writ of certiorari.") Since Brown is a Supreme Court case, we can guess that Brown lost in the lower court; and indeed that's what happened in this case.
There are some arcane rules about abbreviating case names. The full name of this case is "Brown et al. v. Board of Education of Topeka." If you're a law student, especially if you're on the law review, you get to learn all about these rules. If you're not, thank your lucky stars and try not to worry about it.
Volume, reporter, and page number
347 U.S. 483
This is the bit you want to copy and paste to find the case. It's a lot like a URL: the idea is to clearly identify and locate a case. If you just want to find the case on the Internet, you don't need to understand how it works, and you can skip to the next section.
These three parts tell you where to find the beginning of the case in a good old-fashioned printed book. You know those yellow books in the background of lawyer advertisements? That's what we're talking about. They're called "reporters," because they originally contained somebody else's reports of what judges said in court. Now, of course, the judges write down their own decisions; but the name stuck.
The middle part, here "U.S.", identifies what set of books the case is in. You can learn something about the court from this:
U.S. -- United States Reports -- United States Supreme Court
F., F.2d, F.3d -- Federal Reporter -- Federal Courts of Appeal
F. Supp -- Federal Supplement -- Federal District Courts (trial courts)
The first number is what volume the case is in. This is the number printed on the spine of the book. The second number is what page the case begins on.
When somebody wants to refer to a specific place in a case, they just put the page number or range right after the case page number. Remember, "483" was the page the case starts on; "495" is the page where the court actually say what we're talking about. If you flip to page 495 of volume 347 of the U.S. Reports, you'll find the quoted sentence.
Remember: usually when you see two numbers separated by a comma, like "483, 495", the first one is part of the citation to the case as a whole, and the second one says what page to look at.
Date and court name
This part varies a lot between the different formal styles. In "Bluebook" style, which most law journals use, the year the case was decided is enclosed in parentheses, along with anything necessary to identify the court. Here, there's no court name, because we already know from the "U.S." that it's a Supreme Court case. For lower appellate cases, this might be "(9th Cir. 2005)", meaning that it's a decision of the Ninth Circuit Court of Appeals from 2005.
Dealing with short citations
You may also see something like "Id. at 495". This is a short citation; "Id." means "the last thing I cited." Go back to the nearest previous citation and look that up, going to the page cited.
How to Use It (aka "tl;dr")
The key to actually finding a case on the Internet is to copy the three numbers in the middle--the volume, reporter name, and page number. Here, "347 U.S. 483". Don't include any pincite after a comma; Google and many search engines may be clever enough to figure out what you mean, but they may not.
There are a bunch of free online services providing case texts. I prefer Google Scholar; just paste the citation in, being sure to select "case law". Other people like CourtListener, FindLaw, and Justia.com. Commercial databases like Westlaw or LexisNexis have a lot more than just the case; most importantly, they list citations to the case. Your local public law library might have a terminal for one of the services that you can walk in and use.
Be sure you're reading the actual case, not a summary or "annotation." This is especially true if you're writing an answer here on Law.SE. Many case summaries, especially for famous cases, are written for first-year law students; some of them are probably written by first-year law students! They're often helpful, but they focus on what the student needs to know for class, which often isn't what the case actually said. If I find you quoting a case summary as if it's "the law," I will not be very nice to you.