MantisBT is available in several Linux distributions including: Debian, Ubuntu, Fedora, Gentoo, Frugalware and others. Hence, if you are running Linux, start by checking if your distribution has a package for MantisBT. If not, or if the package is not up-to-date with the latest MantisBT version, then you may want to download it directly from here.
For Windows, Mac OS X and other operating systems, use the link provided above to download MantisBT. The download is compressed in tar.gz or zip format. Both formats can be unpacked using tools like 7-Zip (in case of Windows).
Note that at any point in time there are typically two "latest" MantisBT releases that are available for download. The latest production release (stable), and the latest development release which can be an alpha or a release candidate. It is not recommended to use development releases in production specially if it is still in the alpha stage unless the administrator is familiar with PHP and is able to troubleshoot and fix any issues that may arise.