LED Driver IC STLED316S
My guess would be this:

To switch the display on:


STB goes low
Clock in the byte 00001101 (0x0D)
STB goes high


To write the segments:


STB goes low
clock in the Write command: 00000000 (8 bits of zeros). This means: it's a write command, with auto-incrementing addresses, writing to address 0x00 on page 0x00.
Now you can clock in the data to fill the memory. This consists of six bytes, one for each digit.
STB goes high


To clock data (make sure the clock is slower than 1MHz):


STB goes low
Loop for 8 bits of the byte

Set data out bit
Clock goes low
Clock goes high

End of Loop
STB goes high