Although the terminology used in this question is common in popular science articles and books, it is clearly misleading. The term “awareness” is not something electrons can be aware of in the same way that humans do.
This type of sentence usually refers to a subtle aspect of quantum Physics that, at the moment, does not have a consensus resolution. This is the famous issue of Interpretations of Quantum Physics. They attempt to explain the counterintuitive aspects of quantum physics by using different fundamental frameworks.
An example of what the phrase “electrons are aware they are being watched” may be the fact that electrons behave differently in double-slit experiments when one or both of the slits are closed. The slits in a lab experiment are only a few centimeters apart. However, the experimental principle is applicable to large scales. We start to see the problem when we look at slits that are more than a meter apart. When they are spatially separated, how does an electron passing through one of the openings “know” which slit is open and shut?
A single electron that is shot through a double-slit experiment will behave differently depending upon the distance between them. What does an electron passing through one of the slits learn about the distance between the two slits?
Accepting that electrons travel simultaneously through both slits and through all available paths, is the only way to provide a satisfactory solution.
This does not mean electrons are “observed”, but they “know” what their space environment is. They know if there is one or more paths they can follow or 50 or infinite paths. And they will act accordingly.