Your washing machine could be sending 3.7 GB of data a day — LG washing machine owner disconnected his device from Wi-Fi after noticing excessive outgoing daily data traffic
If you have anything where internet is an add-on to what it does normally, especially BS like a washing machine, then it's phoning home. That's the reason they add such nonsense, and sell it as a feature to the buyer.
They have to run a backend for this stuff which eats into the profit of selling it...
That said, Wireshark is a common tool to monitor packets. I haven't done it for a while. There's also probably a package you can run on RPi just for this kind of thing. Using PiHole I can see how often and where devices are connecting. I've blocked a lot of domains - I'm currently blocking about 30% of all domain requests (most of that is from the TV and windows 10) and everything works fine.
The software would need to be on the router itself so that it can intercept all traffic that is originating from the LAN (Local Area Network) and is directed at the WAN (Wide Area Network, the Internet), some higher end (home) routers have this feature, or you could see if your router supports a third-party firmware and flash that, which most likely supports it.
The other (more complex) way is to put a device in between your router's LAN port (usually called the Default Gateway in software terms) and the rest of your devices on the network so that it intercepts all the traffic and then forwards it to the router. This is a technique known as "Man in the Middle".
If you want to simply know if devices are "phoning home", and potentially blocking those requests, you can use a service like NextDNS or PiHole (on a Raspberry Pi, or in Docker) to block the attempt. This happens because the device doesn't know where to send the info.