I am using SonarLint plugin for IntelliJ and when setting this up it gives me the option to select a SonarQube project from the list of all SonarQube projects available as you can see in the picture below
What exactly does this do when I select a particular project as I am not exactly clear on this and it does not explain on the SonarLint main site?
Does it take the rules for the selected project and use these rules for any project I am working on in IntelliJ?
This option sets up "Connected Mode" so that, as you guessed, the rule set applied to your project matches the one on your server.
You explicitly connect a single workspace project to a single SonarQube project. The default rule set(s) will be used on un-connected projects.