This article is from the Apple II Csa2 FAQ, by Jeff Hurlburt with numerous contributions by others.
A "TV/Game" switch is, usually, intended to switch RF (radio frequency) signals. For example, in the "TV" position it connects the TV's VHF antenna input to the VHF antenna (e.g. the long "rabbit ears"). This way, you can watch TV. In the "Game" position it connects the TV's VHF antenna input to a game machine's or computer's CH 3 or CH 4 output. To see the computer's display, you need to set the TV Channel selector to the channel being output by the computer. The reason your setup does not work is that the IIe does not output an RF signal. (In fact, no Apple II has built-in circuitry for outputting an RF signal.) The IIe outputs composite video. Composite video is pure video information like the kind output by a VCR through the standard 'Yellow' cable. It produces a better, sharper image than video which is converted to RF and fed into a TV's antenna input. The usual way to connect an Apple II to a display is described in Q&A 018. Getting an "RF Converter" ("TV Converter", etc.) module is another way to go. For about $30 Radio Shack sells a box which will convert an Apple II's video signal to RF on CH3 or CH4. One thing to consider when looking for ways to connect to a color TV without a standard "Video" input is that you could end up wasting time and money better spent on just getting a standard composite color monitor (like the Amdek Color-1) at a local Apple II swap meet. ----------------------------