Basically there are two ways to download SilverStripe modules from addons.
1. The 'oldfashioned way' by downloading the source code from the GitHub repository in a Zip file. In this case the 'Homepage' link on the addon page will lead you there: https://github.com/silverstripe-labs/silverstripe-sqlite3.git. To the bottom right you will find the download button.
Note: an older version of SilverStripe might need a different version/branch of the module. Check the Readme documentation and if necessary use the top left button for the right branch before you download.
2. The 'modern way', by using Composer - software that will download the module for you based on a composer.json configuration file. If you opt for composer, you can scroll down the addons page and copy the 'require' reference for the correct branch/version to add to your composer.json file. For information on how to use composer checkout the docs here: http://doc.silverstripe.org/framework/en/topics/modules#installation
All this might look a bit complex, but in the long run investing in Composer pays of, because once you create your own set of composer.json files, it's a breeze to build custom setups using the newest updates, and install in a wink 
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