You have the right datasheet.. the one I linked does have different pin connections.
It looks like your connections to the LED display are mostly correct. It looks like you've partially wired the connections for the colon between the digits (pins 7 and 8 ), but have left the wires floating.
Your connections to the '595s need a bit of work though.
The first (and probably most frustrating) issue is that your Output Enable pins (pin 13) are floating. You need to connect those to GND before anything will show up on the data pins. While you're in the neighborhood, connect both Memory Reset pins (pin 10) to VCC. That will keep a random voltage spike from erasing the data you send.
You also have the two chips connected in parallel -- they both get the same data and clock signals -- and what you want is a serial connection. To make the description easier, I'm going to call the chip on the left (that controls the individual segments) C1 and the chip on the right (that controls the transistors) C2. I'm also going to use a name.pin notation, so 'C1.13' would be the Output Enable pin for the device on the left.
Leave C1.14 (the first chip's data line) connected to the Arduino, but connect C2.14 to C1.9.
The '595 has an internal shift buffer and a 'latched buffer' that connects to the output pins. Every time the data clock signal goes HIGH, the chip moves everything in the shift register one place higher and copies the signal at the data pin into the lowest position in the shift register. When the output clock signal goes HIGH, the chip copies everything in the shift register to the output buffer, with the newest bit appearing at Q0 (pin 15) and the oldest bit appearing at Q7 (pin 7).
Pin 9 is connected to the last bit in the shift register. It shows the value that will go to Q7 if you send the output clock signal HIGH.
Connecting C1.9 to C2.14 hooks the two shift registers together. When the data clock goes HIGH, the oldest value shifted into C1 becomes the newest value shifted into C2.
So.. to drive the display, you need to send 16 bits. The first 8 will end up in C2, and the last 8 will end up in C1.
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