Without a schematic it's difficult to tell exactly but I would guess from the picture and the text on it that all you have to do is send it the correct sequences of signals. The decoding and driving circuits seem to be there already.
Firstly, it's an LED display, not an LCD!
The 2 to 4 decoder seems to be connected to transistor drivers and then on to the common connections of the LEDs, This suggests that you have two incoming signals and the combinations of 00, 01, 10 and 11 on them will enable the display on one of the LEDs. Each combination making one LED light up and the others turn off.
The SIPO and the array of resistors above it suggests the segments of the LEDs (individual bars that make up the digit shapes) are sent in serial. A string on 1s and 0s sent, one after the other to make the appropriate segments light up. There will almost certainly be a clock signal to synchronize the incoming bits with the shift through the SIPO.
So my guess is the way to drive it is this: Starting with the first digit, you clock the pattern of segments into the SIPO then send the signal to the 2-4 decoder to activate the first LED. Then you repeat this for the remaining three LEDs and then start over agin from the first one. Only one LED will be lit at a time but if you do it fast enough, your persistence of vision will make it appear that all the digits are there all the time.
Without a schematic or at least a list of external connections, it isn't possible to confirm all this but I'm fairly certain that's what you have.
Brian.