What does this mean and how do we do it?
A few personal thoughts/observations:
Awareness is not paranoia or hypervigilance. Going through life expecting trouble around every corner is psychologically damaging. It entails living in a near constant adrenalized state, which would appear similar to an anxiety disorder. Such a state may be appropriate for soldiers outside the wire in a war zone, or a cop walking an extremely rough beat, but it's not necessary for the average citizen and not conducive to a long, happy, healthy life (which is kinda the point of self-defense).
Instead, think of Awareness as simply being engaged in the real world around you. Many people today go through their daily lives primarily inside their own heads or surrogate heads (electronic devices). They are so out of touch with what's actually going on around them that they'll unwittingly bump into people and objects, walk into moving traffic, miss turns (or worse) while driving, and generally not notice anything that's going on around them. That's both dangerous and kinda sad.
Believe it or not, there are lots of beautiful, strange and wonderful things going on around you nearly every moment! Most people miss them. If you engage with and observe the real world around you, without any thought of looking for or expecting trouble, chances are good that your subconscious mind WILL pick up on anything that seems wrong, out of place or dangerous (e.g. someone following you, someone loitering, etc.). You must trust your gut (intuition), because often you'll "feel" something is amiss before you're consciously aware of what it is. If you ignore that feeling, you may stumble unknowingly into the lion's jaws. Trust your gut and act on it.
Save your ruminating and electronic gadgetry for secure locations and circumstances.
How can we remind ourselves to keep our attention outward? Figure out some mental "games" you can play that are enjoyable. You could, for example, try to count how many red cars or men with beards you see at any moment . . . but to me, that's boring and therefore I won't usually do it. The most obvious game, that many folks do naturally and which IS decidedly enjoyable, is called "Look for Hotties." Wherever you are at any given moment, challenge yourself to always find the most attractive people (or vehicles, or animals, or whatever you're into) in your immediate vicinity. This should keep you happily engaged with everything around you in your quest for hotness. Just remember not to "lock onto" any particular subject, because it's rude and it shuts down your 360 degree awareness in favor of tunnel vision. Just note all the good looking specimens and keep scanning for something even better. In the process, you'll see what you need to see to keep yourself safe.
Depending on your circumstances, you may not want to explain your awareness strategy to your significant other in detailed terms--but certainly suggest that s/he give it a try, secure as you are in your relationship! (NOTE This blog does not purport to offer relationship advice. Results may vary.)