After purchasing or signing up for the free trail, you will have received an email with a download link.
Us the download link to get the plugin file and then install via your WordPress admin dashboard by going to Plugins>Add New>Upload Plugin
Activate the plugin, this should automatically deactivate teh free version.
If you check your installed plugins you should now have
Display Eventbrite Events in WordPress (Pro) – Active
Display Eventbrite Events in WordPress – Inactive, which you can delete.