Brave browser connecting to other devices in my network

From a cursory scan, these appear to be Fastly IP addresses (which we use for our service endpoints) and other Brave-related pings. For instance, I see one of these is likely associated with our Discourse instance here at community.brave.com.

The calls to port 8009 on various local addresses appears to be Chrome Cast Detection; the browser is attempting to locate devices on your network which are able to receive a media stream.

If you want to be absolutely positive, you could download Telerik Fiddler (works with HTTPS too, via a local root cert), and monitor the activity with that app. That will show us host names for these connections, very likely revealing what’s going on here.