Google added support for WiFi communication to Android Wear in the 5.1 update a few months ago, but not all watches had the necessary hardware. The pricey LG G Watch R was among those lacking proper support. LG said it planned to rectify that in an update over the summer, and indeed it has in the new Wear update.
Having WiFi on Android Wear means your watch can communicate with the connected phone over WiFi instead of a direct Bluetooth connection. As long as both devices have an internet connection, they will remain paired. Devices like the Smartwatch 3 and LG Watch Urbane had WiFi support as soon as it was available, but LG needed to do some tinkering to make WiFi functional on the G Watch R.
The new version of Wear will be delivered to all watches via OTA update in the coming weeks. If the OTA URLs are captured, you can bet we'll post them here so you can sideload.
- Android Developers