The strange thing is that on the side of my remote TV, the LED on the pyramid goes on when the TV in that room is on. There's no physical connection from the TV to the pyramid. How the heck does it know that the TV is on?

It doesn't. It's not directly reacting to the TV, but to environmental conditions (perhaps indirectly related to turning on the TV).

I didn't catch clearly which of the two modules is the one that lights up. But here is what is supposed to light the LED:

- On the transmitter module (the one without the rabbit-ear antenna), the LED will light up when it sees some kind of infrared light through its sensor. For instance, on the one that sits on my kitchen windowsill, the LED light blinks randomly pretty much all the time during sunny days (although I got that filter material from David yesterday and I have high hopes it will fix the problem).

- On the receiver module (the one with the rabbit-ear antenna), the LED will light up when it receives a radio signal anywhere near its frequency range that it thinks is a transmitted IR signal. This could be RF interference from other electric/electronic devices (perhaps the TV itself), or it could be receiving random IR signals from the transmitter module.

In both cases, maybe altering the placement of the module will change the response.