watchOS 3.1.3 can be downloaded through the dedicated Apple Watch app on the iPhone by going to General --> Software Update. To install the update, the Apple Watch must have 50 percent battery, it must be placed on the charger, and it must be in range of the iPhone. watchOS 3.1.3 requires an iPhone running iOS 10 to install.
We don't know yet what new features or bug fixes might be included in the watchOS 3.1.3 update, but no outward-facing changes were found in the first beta. watchOS betas are often rather minor in scale, focusing on small bug fixes and performance improvements rather than major interface changes.
The second watchOS 3.1.3 beta comes after Apple pulled the watchOS 3.1.1 release after it was found to be bricking some Apple Watch Series 2 devices. Apple has not re-released an updated version of watchOS 3.1.1 that fixes the bug, so many users are still running watchOS 3.1.