How is the common anode 7-segment display in Ben Eater's setup working?
Your mental model needs updating.


  The way I understand common anode to work is that a given segment of the display will only light up if it has a path to ground.


An LED (by itself or inside a display) will only light up if it has current flowing through it.  This will happen if it has voltage across it.  One way of making this happen is to hold one end high and give the other a "path to ground" -- but that's not the only way, by any means.

In the common anode configuration, the anode is at VCC.  When a cathode is pulled to ground through a resistor, current flows and the LED lights.  When a cathode is "pulled" to VCC through a resistor, there is no voltage across the LED/resistor combination.  No current flows, no light lights.