I have a home Appliance that has a digital LED readout on the front showing
either temperature (good) or error code (bad).  

I would like to build a device that hooks up to the Appliance so that the
device--presumably based on an arduino-like microcontroller--can read the
signal that is otherwise sent to the LED display and send the display
information or some digitized version of it over a wi-fi network so it can
be polled remotely, all without disrupting the original Appliance's
operation.  I have access to the circuitry and can read numbers on chips
but I don't know how to go about analyzing the circuit to see where the
signal lives.  I would appreciate any help or advice! Thank you.
-Michael