Just like it reads.
The Semantic Web is an evolving extension of the World Wide Web in which the semantics of information and services on the web is defined, making it possible for the web to understand and satisfy the requests of people and machines to use the web content. It derives from World Wide Web Consortium director Sir Tim Berners-Lee's vision of the Web as a universal medium for data, information, and knowledge exchange.
The simplest and shortest explanation that I have found is: "The Semantic Web is to Machines what the World Wide Web is to Humans".
And as to why you would want that: for the same reasons why you let your Machine compute Pi to the quadrillionth digit instead of doing that yourself. So you can focus on interesting problems and leave the menial work to the Machine.
Well, it might not be fitting in with the "official" definition, but I always try and explain it to people as "It's like syndicating knowledge instead of content."
Why would you want to use it? Well... if you are making applications that could benefit from machine parseable and queryable "knowledge," then... you might want to use it :).
IMHO it's rather ill-defined and not implemented in a broadly useful way at present. It's a good idea and I have no doubt things will tend towards this sort of approach in the future, but it's not there yet.