Gamesense is the art of completely understanding your opponents movements and actions, to the point that it is near-impossible to play against you. The reason good players like star_ and jerma are able to survive for a long time and wipe out whole teams is because their gamesense is very powerful.

In general, the only way to improve your gamesense is to just play a lot. Fortunately I can still give you tips to improve faster.

The class best suited for gamesense is the pyro, hands down. The pyro's flanking nature and airblast skill both heavily rely on gamesense. Next time you play pyro, do two things to help improve your gamesense:
1: Figure out where your enemy is going. This is done with a combination of the objective, target, and current position of the target. If there is a heavy on pl_upward with the gloves of running urgently moving towards the cart, chances are he will go straight to it. Use this knowledge to change your path accordingly to help you get the jump on the heavy.
2: Airblast at very close ranges. This is intended to reflect projectiles, rather than knock your opponent back. Eventually your gamesense will get to the point that you can reflect almost all projectiles the instant they are fired. This is the anticipation part of gamesense, where you learn how and when an opponent will fire a weapon. Often this relies on feeling an opponent's reaction time, and knowing if he will spam-fire or put thought into when he will fire a projectile. Eventually this skill can be applied to other classes, helping you to avoid projectiles such as pipe bombs, rockets and huntsman arrows.

Eventually, your opponents actions should become something that you anticipate intuitively, requiring no thought, merely action.

When you think of gamesense, the scout is one of the last classes to come to mind. What you don't know is that the scout relies heaviest on gamesense, even more so than the pyro or soldier.

The reason for this is that the scout class is best played as a class that attempts to avoid damage. Knowing where an enemy is and what he will do is crucial to this mindset.

It becomes easy to see this from a simple scenario. Scout vs. Heavy creates almost a perfect scenario to see how vital gamesense is too have.

1. The scout approaches the heavy. A good scout will have a good ambush oppurtunity, but that will only be good for an extra two shots on him.
2. The heavy revs up his minigun. The scout sees this, and prepares to deal with the oncoming spray of bullets.
3. The scout zig-zags in wide arcs particularly avoiding the minigun fire. The scout also takes notes on when the heavy begins looking the other way to determine how often he should switch directions. While zig-zagging, the scout also jumps when the minigun fire passes by him, allowing him to bypass even more damage.
4. The scout completely closes the distance between him and the heavy, and takes out the heavy with a few well placed meatshots.

In this scenario, the scout had a powerful gamesense, and used it to his full advantage in dealing with the heavy. In knowing how the heavy will move and aim allowed the scout to avoid most if not all of the damage. A normally incredibly powerful high damage dealing class is reduced to a heap on the ground, having achieved little, decimated by the awe-inspiring power of gamesense.

You have reached the end of my guide. I know it was short; I am sorry. Unfortunately, there is only so much that can be "taught" about gamesense. It needs to be largely learned, and the only way to do that is with practice. All I can do is guide that practice and tell you what to look for. I hope that this guide will help you become a better player, who plays smarter. The next time you turn on TF2, I hope that you will play like a god.

Thanks for reading!