To install Jekyll and build: (Note for windows see: https://stackoverflow.com/a/16853614/1596547)
- install rubygems (sudo
apt-get install rubygems)
- (sudo)
gem install jekyll
go to the ./docs dir of your bootstrap install (see also: Explore and install Twitter Bootstrap 3)
- run
jekyll --server from your bootstrap dir
- visit
http://localhost:9001/ http://localhost:9001/docs.html in your browser
NB the default port for WEBrick will be 9001. If this port is not avaible an other port will be used: WARN TCPServer Error: Address already in use - bind(2)
INFO WEBrick::HTTPServer#start: pid=30728 port=4000
When you visit http://localhost:9001/ you will find a home screen. The "View docs" button links to /docs/ while the docs are on docs.html See below:

Now (jul 10 2013) the navbar links seems broken (missing .html) so use http://localhost:9001/javascript.html in stead of http://localhost:9001/javascript. See also: https://github.com/twitter/bootstrap/pull/8443
Using git (jekyll already installed):
git clone --branch 3.0.0-wip git://github.com/twitter/bootstrap.git
cd bootstrap
jekyll --server
visit http://localhost:9001/ or http://localhost:4000/
UPDATE @lee-whitney suggests to use jekyll serve instead of jekyll --server. The jekyll serve command creates a directory with the html files of the docs. You could serve this files on a (local) webserver. jekyll serve don't start a (jekyll) server. I test this commands on linux (Ubuntu 12.04LTS) with Jekyll 0.12.1