how can NAS work if it is not connected to Internet?

The Ethernet ports in your router are just a switch, basically. The home network does not need internet to work internally. The router will still assign IP addresses, if the option is enabled and the network will still work. You just cannot connect to the internet or to your router from the internet. This is why a NAS like this will still work, but only on your internal network, which means only via WiFi provided by said router and from devices connected via Ethernet (LAN) cable. No access from the internet will be possible. This includes phones not connected to the internal WiFi.

NAS means Network Attached Storage, which means, that the device is, in it's essence, a small computer with a connected hard drive. So all the storage remains locally and is, normally, not uploaded to the Internet, unless configured otherwise. The NAS is sharing access to the device via network folders and, most commonly, uses the SMB Protocol. Most NAS also provide other means of access, such as File Transfer Protocol (FTP) and various others.

If this NAS is not allowing usage without Internet Access, then it is a serious flaw that should not exist and flat out a bad product. It is essentially doing what is commonly called "phoning home".

According to the description (and the title), this isn't just a NAS. This is an internet cloud storage device, and part (most?) of its job function is to allow you to access whatever's stored on this device remotely - i.e. away from home.

Cloud storage is (roughly) defined as storage areas accessible as long as you have an internet connection. If you turn off the internet access to this device, then you won't have access to any of the files outside of your home network. One thing to note - if certain "standard" tools are required to access files, and those "standard" tools require the internet, you may need to find alternate ways or apps to access the device. If there are no alternative ways to access the device on your home network without internet access, then I would consider the device to be "defective."

The cloud functions of this device would (and should, if you feel you need these functions) require access to the internet. If -- and this is a big if -- the primary / sole purpose of this device is for cloud storage, then it might not be a NAS at all in the traditional sense. If that's the case and you don't need any cloud storage features, then I would look elsewhere.

Hope this helps!

I would expect it to depend on what software you use to do your backups.

Having a NAS of a different type I have at least 3 different ways to access and send data to the NAS.

1. I can go to the web-server on the NAS, log in and upload files that way
2. I can connect via SAMBA/SMB and browse the filesystem like any networked computer using the \\MyNAS\a\folder\on\the\NAS\ notation. This is also how the Windows backup and File History features access the NAS to back up your data.
3. Media server software - dependant on how that software works
4. Particularly for mobile - The supplier provided discovery and backup tools.

I would expect that the users problem was with item 4 on that list. In particular I found that Western Digital software automatically tries to make you create an account to link your NAS "through the cloud" so that you can access it anywhere. Some of us don't want that and disabling that level of featuritis can be non-trivial in some cases. They don't tend to make it impossible though and you should be able to use their tools only within your own network.

By no means do you have to use their software to do things, but it may just make things smoother.