Podcast players like Overcast use push notifications to learn about new episodes of podcasts that should be downloaded in the background. Presumably text-based RSS readers do the same.
Where are the push notifications originating from? Does Overcast have a cloud service that polls the RSS feeds and then sends the notification? I use AntennaPod on Android, and it definitely doesn't do anything like that -- the feed list is stored locally, and the feeds are polled locally.