One typical example for this feature would be to trigger new build from the source control system's hook script, when somebody has just committed a change into the repository, or from a script that parses your source control email notifications.
You'll need to provide an authorization token in the form of a string so that only those who know it would be able to remotely trigger this project's builds.
This is most useful when your Jenkins instance grants read access to this job to anonymous users.
When that's not the case, Jenkins will reject requests sent to the trigger URL even when the correct token is specified.
To solve this, the HTTP requests needs to be authenticated as a user with the necessary read permission for the job — but then you could probably just grant this user the permission to build this anyway.
Another option is to use the Build Token Root Plugin, that provides additional URL endpoints to trigger builds using this token, and doesn't require the otherwise necessary Overall/Read and Job/Read permissions to do so.