Tag the commit you want to release as a version in your Git repo, push the tags to Github. Works on every repo as advertised. It might be a good idea to stick to semantic versioning, so make a version number from three parts: 1.0.0, and increment them according to your changes.
Enable the post-push hook on Github to alert Packagist when you pushed - otherwise there will be more delay on Packagist to see the new versions.
And that should be it.
Please make sure you instruct your users to require a tagged version, do not advise them to use "dev-master". And tag a new version as soon as possible if you collected a substantial amount of new features or bug fixes.