A good real world example is when you're walking as a pedestrian. You have to keep an eye out and can't just walk with your eyes closed. The same is true if you are driving or are doing any kind of activity where you need to concentrate. In a game such as BW where you are faced with many tasks that have to be accomplished in a finite period of time, awareness becomes extremely important to master in order to become a better player overall.
Infact, I would even go as far as saying that you will fail as a player if you lack good awareness.
Sound harsh? Sure, but I personally believe it's a true statement.
Let's take a simple early game TvZ situation where not a lot of things are going on. You as the terran have already went for a 1 rax FE getting the cc on the low ground on a map such as Circuit Breaker (long map for any non-BW folks). You scout the zerg who spawns close position. Unfortunately, you are a noob and lose your scouting SCV after seeing that his pool is nearly completed. With the SCV gone, you have no further information to go off and are playing in the dark. You become aware of your current predicament, switching from the usual autopilot that usually grips you after you've played enough games.
You have several options and they depend on knowledge of your opponent's playstyle as well as the weakness of your build. It also depends on just how risky you want to play:
1) Send out an SCV as soon as you realize that you're going to lose it to keep scouting.
2) Get a bunker and play it safe. This is advisable if your opponent is a zergling abuser! lol.
3) Forgo the bunker and just park some marines behind an SCV on hold command.
You might be thinking cmon Bigfan, the logical thing to do would be #1. Oh, I would agree! I think #1 is a fine option as well however, there are several things to consider:
1) Can this SCV survive enough to catch any hint of what the zerg is doing?
2) Depending on your build, that SCV might be worth too much to have it scout around.
Keep in mind that the game is also continuing as we speak so you can't just ask the opponent to pause and give you a chance to consider your opponent. As the game continues and you are forced into making a decision, you have to keep maintaining awareness of your situation at home as well. You have to keep a constant eye on your resource and supply count as well as keep an eye on the map. I've lost count of how games I've lost when I first started playing 1v1 due to having zerglings slip past my bunker when my attention was diverted elsewhere. You might say this comes down to multitasking as well. Sure, these ideas intertwine anyways.
You decide to go with option 1 reasoning that it's fine overall and worth the income lost. Your SCV miraculously makes it's way to the opponent's base and you manage to make it up the ramp only to run into 4 zerglings. The SCV survives just enough for you to see drones pop out of the main hatchery. This makes you feel fairly safe and you decide to forgo the bunker instead parking a single SCV in a narrow choke while you continue marine production. Should you just relax and assume that your opponent made 2 extra lings for the sake of denying scouting?
I'd argue that this is the point where you are most vulnerable. Before you scout the drones, it was a toss up, but now you've let your guard down and guess what happens? You get zergling busted. Turns out your opponent had better map awareness than you did and spotted the incoming SCV (see video for map awareness example). Realizing that it'll make it into his base, he made some drones in the main, but kept pumping zerglings at the other 2 hatcheries. Sure, this is mind games now, but you can't deny that you could've done more to catch on? Counting drones at his expansion or keeping the worker alive or sending out another worker to keep scouting around the entrance of his base. You need information after all!
Let's kick things up a notch!
Let's assume that you were correct and your opponent goofed up. He meant to make 2 lings, but made 4 otherwise he was just droning and was playing a standard game. Fast forward to a bit later after you've added gotten an acad, added some rax and have stim researched. You are getting close to have your first mnm group out. You've already gotten a comsat station and scan your opponent to reveal both a spire and a hydra den. You move out towards their base when you're ready, but you're in a bit of trouble too since you're unsure of what your opponent opted for. What do you do?
For starters, scouting is always key. If your opponent decided to go for a lurker bust, there'll be signs of it. Let's say they went mutas and you've caught on due to seeing a muta, but how can you be sure that they didn't also get a lurker or two? Hold lurkers can be a pretty big threat in BW games. Savior was lauded for his accomplishments during his bonjwa years precisely because he knew how to play the zerg race with smart positioning, threat of counter attacks, isolating armies and cutting off retreats/reinforcement etc... Some of this stuff might be more standard now, but it wasn't at the time from my limited understanding.
Maybe the opponent went for the 7 muta 2 lurker build? Typically, without being able to detect lurkers, it's not advisable to move out. You count the mutas and find out that he's made more than 7 of them so you decide that it's safe to at least leave your expansion. You move towards the direction of the third base, but as you get near the bridges, your spidey sense is tingling. Several options present themselves:
1) Do you walk across the bridge with your army?
2) Do you go up on the high ground and make your way around the bridge?
3) Should you just ditch and go home?
If you can't confirm 100% that his lurkers are not on the other side, the smart thing to do would be to turn around and go back home. However, let's not forget that you have some scans available now. If you want to save them, the alternate is to send a single marine ahead to scout. If he gets skewered to death, you know that he lost his life for a good cause. You can predict how many there are and the position of the lurkers from where the spikes originated. Just another example of being aware of the situation.
Let's take things to the extreme end!
It's now late late late game TvZ. One of the most difficult tasks in the late game as terran is trying to predict where the opponent will strike, with how many units and how much units is enough to keep around for defense. Basically, a good exercise in space control. While a good defense is a great offense, you can't leave your base naked and open to a counter attack. More than that though, in the late game, you are now facing the brunt of it all:
A) Have to manage 4+ mining bases!
B) Have to keep macroing, while keeping a close eye on resource and supply count!
C) Have to keep tabs on their expansions/expansion paths
D) Have to stay aggressive otherwise the zerg will take over the map
E) Have to predict where the opponent will strike through scans and prepare appropriately
F) Have to keep tabs on their unit composition and adjust as you see fit
G) Have to watch the minimap closely
All of this requires some form of awareness:
A) By looking at your counts, you can get a good, but imperfect idea of how well you are doing macro-wise prompting you to add more prod structures etc...
B) By scanning constantly and keeping units at certain areas of the map (think bases or base entrances etc...), you can react accordingly and change your plans on the fly. This is extremely important because late game zerg has an OP unit called a defiler! If the zerg manages to land a dark swarm near your base entrance, unless you have enough melee units and ones that can do splash, it can cause crippling damage, if not end the game right then and there (you can see the videos below for details). By keeping an eye out for attacks, you can irradiate defilers well in advance and increase your chance at defending. In some cases, the zerg will just withdraw and not bother attacking.
C) By studying their unit composition, you can adjust yours to maximize its effectiveness. Also, another small tip is that you can monitor their upgrades and in turn that can help remind you to upgrade your units as well if you forgot.
D) By keeping an eye on the minimap, you can detect unit movement and save scans for when you really need them (your vessels were sniped and 2-3 lurkers are stopping your whole army! lol). You can also catch map pings which can help you get a ton better at macroing. Alternatively, you can also hear the unit sounds too in BW, but you need good awareness for either one since your can get really absorbed by the game.
I could write a lot more, but I feel that this is long enough already. To conclude, for a player to succeed or at least improve in Starcraft, they need to work on a lot of different things with awareness being one of the most important ones. For anyone who wants to increase their awareness, you can do several exercises in real life such as deciding to keep an eye out for a certain object (ex. blue car) daily etc...