When this is turned on in Settings > Advanced Features, a persistent notification will be shown in the notification drawer that allows you to open a pop-up window to compose a new text message. This allows you to start or reply to a conversation without leaving your current app. You can also access this quick compose dialog from the Quick Settings tile (on Android 7.0+) and the launcher shortcut (on Android 7.1+).
Pulse SMS' quick compose is also powerful. It allows you to specify up to three favorite contacts that will be added as buttons to the Quick Compose Notification. Tapping any of those notification buttons will display the quick compose dialog with that contact pre-populated. The favorite users in this section are unique to the quick compose section and are not used in any other portions of the app.