737/800: is the type of aircraft and specific model (Boeing 737, stretched version (800)).
Box-IFE-ICE-SATCOM: Is a theoretical (or actual?) exploitation path.
Box: I'm assuming is in-flight WiFi
IFE: Is the in-flight entertainment system
ICE: Is also part of the IFE, but I'm guessing he specifically referenced that due to the "I" (in ICE) namely, the information that gets fed into the IFE from the flight systems (speed, altitude, and position)
SATCOM: The uplink used by in-flight WiFi but also the IFE to provide "latest news." If can be used to deliver information to the airline about the aircraft so they can keep track of if its on schedule and such.
EICAS messages: Used on aircraft's secure network for flight crew alerts and diagnostic information. Some of these may be relayed onto the insecure network and forwarded to the airline (similar to ACARS, but over SATCOM).
PASS OXYGEN ON: The implication is he wants to cause the oxygen masks to drop down into the passenger cabin (although in reality sending this wouldn't do that, it would just set off a warning on the flight deck letting the pilots know that the oxygen masks dropped, even if they physically hadn't).
If he could send EICAS messages to the aircraft's secure network, that would be a legitimate safety concern. However if he just witnessed EICAS messages from the insecure network, that isn't really a concern except maybe he could send misleading ones to the airline and give them a metaphorical heart attack.
At worst this guy could send a fake message to other passengers, and maybe cause a IFE system warning in the cockpit. Oxygen Masks are controlled by a completely separate system to the IFE.
* The 737NG has various displays for warnings, but its not quite the same as the Airbus EICAS