Latest released version from CRAN
The recommended place to download the latest release of spatstat is from the spatstat page at CRAN.
Latest released version from GitHub
As an alternative the latest release of spatstat can also be downloaded from the releases section of the spatstat repository at GitHub.
Note that the binaries at GitHub only are for the current release of R running under Windows or OSX Mavericks. (For more choices of binaries go to the aforementioned spatstat page at CRAN).
Between releases of spatstat it may be possible to download a development version. This is a source code only version, which is most easily installed using the R package remotes as explained in the README for the spatstat GitHub repository.