To open the Family page, first open the Organize page, then either double click a family in the main list or click the View Family button, which appears when you hover over it. You will see the open Styles tab, along with the other two, Glyphs and Waterfall.
Styles tab contains: the family name at the top and a toggle to activate/deactivate the entire family, Opentype feature list (hover over it to see the list of features available for the current family, if any) and the list of all the fonts in the current family, including any font duplicates you might have added to FontBase, so you can activate or delete a specific file with the same font.
Glyphs tab has the list of all the glyphs available for the current font, which can be changed using the dropdown menu under the tab name. Clicking a glyph will copy it to clipboard. Please note that without the Awesome features turned on the number of glyphs displayed will be limited to 70.
Waterfall tab is home to a set of sample texts in decreasing font sizes, hence the name. If the font you have currently selected is varible, you will also see a set of corresponding sliders on the left dedicated to font parameters that can be adjusted.