The best way to understand this is you have a router on that creates a network within the home, You also have internet service in your home; but it is not essential for the home network to function UNLESS what you want to function is tied to internet for assist. Like Alexa is for most of it's functions so when your internet is down, you are pretty much missing a good part of the utility of the home network. Basically, when internet is down it is the same as not having it at all.
But, as mentioned using Bluetooth between alexa devices will still function since BT is not dependent on internet. BT is a local wireless signal between local devices, so it works between Alexa devices as long as they are plugged in. So, for example, music sent from phone via BT to an Echo still works.