Would anyone know which Solarwinds service is responsible for the automated nightly backups?
balchana Network Configuration Manager (NCM) is the module that backs up and stores the device configs. You would need to schedule a NCM job to run (nightly, weekly, monthly, etc...). You can configure that job (or many jobs), to backup all devices, or just a subset of devices. You can also further define you device templates to backup command results as well. NCM is an extremely powerful tool, and super easy to use.
Here is a link to the "Jobs" section of the NCM Admin guide.
That should get you started, and hopefully answer you question. If you need anything else, just let us know.