The most basic reason to be conscious of your environment is so you know if and how you need to adjust the way you handle your vehicle. Weather traffic, and lighting conditions should always determine how you drive. Being alert will help you make quicker, smarter decisions in the case of an emergency. If you are on the highway and aware of the car in your blind spot, you will know not to swerve into that lane when the driver in front of you stops short. When leaving a blind driveway in an urban setting, you should be prepared for pedestrians crossing your path. It's not a mystery that we should drive differently through a suburban neighborhood than we would on an interstate through the desert, but we (myself included) don't always do this. Simply knowing what is going on around you and planning your actions based on that knowledge will make you a smarter driver. Clueless drivers will always end up looking like jerks behind the wheel of a car.
In a world of fast food, instant messages, and [fill in the blank] on demand, it's hard to be patient. If you're like me, you plan travel time based on the assumption that you'll be exceeding the posted speed limits most of the time, especially on long highway trips. Plan ahead and leave yourself more time than you think you need for your trip. It's not a race - it never is. A hasty lane change or pass maneuver can be devastating. Trying to beat other cars to the exit ramp, through an intersection, or into a parking lot will probably just make them more aggressive and less patient with you. It may also produce some dirty looks or hand gestures. Overly pushy and aggressive drivers can be really annoying to anyone...even to other aggressive drivers.
Know how to use the road
This may sound silly, but not knowing the rules and etiquette of the road (or just plain not following them) is guaranteed to irritate drivers that do know and follow the rules. Is it going to bug most people if you don't come to a complete stop for 3 seconds at every stop sign? No. Will people get angry at you for going 5 mph over the speed limit? Only a handful. Who will you bother by using the fast lane for cruising instead of passing? EVERYONE behind you. There are situations where neglecting the use of your turn signal can really confuse or annoy others. Knowing how to properly merge into highway traffic will reduce your chances of triggering aggression in others on the highway. It's just like anything else you do in public, use proper etiquette and others will respect you.
If you're a jerk in general, your driving will reflect that, too. And some of you, well, maybe some of you can't be helped. Don't get me wrong - I'm not a perfect driver, but I've found that it's not really that hard to consider the safety and comfort of others while I'm behind the wheel of my own car. Maintain reasonable distance between yourself and other cars. Remember to turn off your high beams as soon as you see the headlights of oncoming traffic. Even at a very long distance, your high beams can blind others. Use your turn signals when you should. Be aware of others trying to change lanes or merge - give them room if you can. If you're not passing anyone, get out of the left lane. Don't cut people off if you don't have to (you don't have to). Other drivers will be more confident in your ability to drive if they observe you being careful and considerate.
Don't be a jerk
Ultimately you don't really want to be noticed by other drivers, unless they are looking for you (i.e. being aware of their surroundings). You may even find yourself getting a rather satisfying wave or head nod of appreciation from other careful drivers. These aren't difficult changes, and a big part of driving is using your common sense. If you don't drive like you're the only one on the road, you'll improve the experience for yourself and everyone else.