If a user goes outside with the Apple Watch Series 3 and keeps their iPhone at home, how will the calling feature work?
The Apple Watch Series 3 GPS + Cellular has LTE built-in. Here are the bands supported by the USA version:
2 (1900 MHz)
5 (850 MHz)
12 (700 MHz)
13 (700c MHz)
17 (700b MHz)
18 (800 MHz)
19 (800 MHz)
25 (1900 MHz)
26 (800 MHz)
41 (TD 2500)
That same page also lists the bands for other countries.
The most relevant part to making the Apple Watch 3 (with Cellular) work without having your iPhone nearby is the eSIM (electronic SIM* or Subscriber Identity Module).
Obviously, having LTE capability is important, but what sets it apart is the eSIM's ability to share your existing wireless account info including the phone number and data plan.
One of the most interesting aspects of the Watch 3’s eSIM is that it’ll share your iPhone’s number. You can think of them as part of the same connection. However, this also means it’ll only work with an iPhone – and that your iPhone and Apple Watch will have to be with the same network provider.
When you have this capability, the Apple Watch appears to the carrier as just another device and when a call comes it, will route it to both the iPhone and Watch.
You also have this ability with respect to data, incoming and outgoing traffic gets authenticated to the network as any other device would which is how you can stream music while you run without carrying your phone.
There is a limitation however. Your device and your carrier must support this technology. You also cannot have a data plan on carrier A and voice/text on carrier B.
* A subscriber identity module or subscriber identification module (SIM) is an integrated circuit that is intended to securely store the international mobile subscriber identity (IMSI) number and its related key, which are used to identify and authenticate subscribers on mobile telephony devices (such as mobile phones and computers). It is also possible to store contact information on many SIM cards. Source: Wikipedia
The user receives the call on their watch since it has LTE hardware built in.
Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.