It is not a very easy concept to explain, to be honest. On one hand it sounds simple, on the other hand even more experienced players can actually lose life and death fights due to false eyes.
When I had to explain it, I thought that this sentence covers most of the concept:
“…cases where there are formations which look like eyes, but in fact they have a defect that allows them to be captured. These are called “false eyes”…”
Along with this example:
This is how I still understand a concept. A “False Eye” is almost an eye, but there is a shape/liberty defect that can cause one or more stones to be set in atari, thus rendering the whole formation dead.