You can find more detailed information about this topic in Contributing chapter of the documentation.
Weblate is being developed on GitHub, you can fork it there or just:
git clone https://github.com/WeblateOrg/weblate.git
This website is stored on GitHub as well:
git clone https://github.com/WeblateOrg/website.git
You can translate Weblate at hosted.weblate.org.