How do I control a seven segment display with a shift register and an Arduino?
The names of the pins are clearly labeled on both the back of the board, and the datasheet. 

PWR_IN is 5v to 15v, 12v preferred. The two regulators on the board provide the appropriate voltages from there.

DIMM IN is the '595's OE, Output Enable Pin
CLK IN is '595's SH_CP AND ST_CP Clock Pins
DATA IN is the '595's SI Serial Input Pin

CLK IN and DIMM IN are buffered to the output side with a buffer chip.
DATA OUT is the '595's Q7", Serial Output Pin, which goes to the next display.
And PWR_IN is directly tied to PWR_Out.

Since Latch and Clock are tied together, a quick search online indicates:
I had to add one more clock up/dw because, as the datasheet says, latch clock is one step behind the shift clock, when tied together as a fix to any issues.

Other than that, this can be driven with the regular Arduino ShiftOut library.

The DIMM IN pin must be driven low/0 for the display to be on.