How do I connect this piece of electronics to show the digit display? (from satellite decoder)
In my experience, the pinout for LED displays of a given configuration and number of pins almost always conforms to defacto industry standards. Here is a similar display available through distribution that I expect would have the same pinout. 

Edit: David Tweed has noted the display is Common Anode, so the below schematic should be correct. The digit-drive transistors appear to be NPN emitter followers. You could retain those and the segment resistors to drive the display from another micro (total 11 lines required). 

This assumes you're going to strip out the display and drive it directly. If you want to continue to use the micro on the board, the protocol is probably not going to be easy to reverse engineer, even if both halves were working.