Far-seeker:
However, I would be sure to measue the voltage outputed by the switch before connecting it to an Arduino. It appears to not have any dedicated power supply. So the voltage levels are likely whatever the manufacturer considered most convenient to get from the HDMI cables, and that could be something either higher or lower than the 5 VDC from an Arduino pin. At worst that means a bit additional level shifting circuitry, but it's something that should be detemined before connecting the Arduino.
That's the point of using an opto-isolator. That way there is no electrical connection between the arduino and hdmi switch (opto-isolator is a chip with a LED and a photo-sensor inside, the arduino turns on the LED, and then the photo-sensor closes the circuit on the other side when it senses light). If it is high voltage, or A/C, you probably want to use a relay instead of opto-isolator.
Otherwise if you connect an Arduino directly to the switch, you need to make sure the grounds are connected, and the voltages are safe, etc.