1) How were early computers compromised like this?
Every electric signal produces an electromagnetic field - aka radio transmission - changing in the same way the signal does. An AM radio near any (old) computer will have a good choice to catch some of it's workings as 'noise'. Of course most radiations as way to broad easy easy detection, but a CRT signal is cyclic and this cycle is high enough to work as it's own carrier - and at the same time delivering the fine print for decoding.
The problems are still the same with modern displays - except the energy is lower and frequency higher, the later even more reducing the distance it can be detected. So an Apple II is way less problematic to capture than a 4k display. A fact even noticeable in demonstrations back then - usually some TV like lower resolution screen was used, than an 80 column or highres display.
2) Were there other ways to compromise the CRT/Monitor signal?
It's a bit unclear what you're asking, isn't it. As usual, anything a setup emits can be detected and used for reconstruction of the signal. The most prominent way here would be the visible picture itself, like in peeking thru a window onto a screen :))
Don't laugh, but it was (and is) used quite a lot. It doesn't need super high tech to spy - a good pair of glasses can do it as well :)
Based thereon, it works as well if the screen is not visible from the outside. For example via reflections by walls. Unlike assumed by humans when viewing a screen, it isn't static, but drawn over and over again. Exactly like the signal is delivered. So targeting the window with a sensible photometer and amplifying the fluctuation will as well enable a reconstruction of the signal shown.
In fact, with CRTs it's not even necessary to have a window, as they as well emit the picture in x-ray band. So at least with paper thin walls and a detector pointed at the screen front it can be captured from a windowless room.
3) Is there a way to eliminate this threat with older computer hardware?
The thread is still there, especially since detection now can uses digital signal processing to filter even better from even fainter signals. It got somewhat reduced due higher resolutions, high frequency transmissions and the use of low intensity displays like LCD.
Counter measures are the same as for all radio and light emissions. shielding for computers, cables and screens (and rooms), no direct sight and no windows will reduce the risk to almost zero - in addition, having several similar devices (screens, computers) in close distance (same room) will scramble up the signal to an almost undetectable smudge.