How does my family get to use the storage plan that I have shared with them?

They use it the same way as they would if not sharing (i.e., they continue to access iCloud via their own Apple ID.) Also see: Share iCloud+ with your family - Apple Support

As long as they are a part of the Family Sharing and you are the Family organizer, when they restart their devices, the amount of iCloud storage should appear on their devices. On your device, go to Settings>Apple ID>Family Sharing, then scroll down under Shared with your Family, and make sure that iCloud+ is there and it should show your Family Members.

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