UDN2981 is used to source the (16 LEDs x up to 20mA) of current for the LEDs in Row1, Row 2, etc up to Row 8.
Typically one would set all the cathodes high/low as desired, then puls the anode current on.
Set the anodes for the next row, pulse the next row's anode current on.
Either the OP has very high current limit resistors to keep the current required per row down, or 'extreme' multiplexing is going on and only 1 cathode at a time is being taken low so only 1 LED can be turned on at any one time, vs up to 16 at a time.