Thanks for doing a search beforehand. It's much appreciated; I hope other posters who don't understand the current model also do the same.
It comes down to what you see is what you get. You see a collision, you collided. The other guy doesn't see a collision, he didn't collide. That's how you die and he doesn't.
The primary thing to understand is that AH uses a smoothing code... your FE is actually predicting where the other plane is going to be based on its last known trajectory/speed. Keep that in mind.
So now we've got 2 planes on a collision course. Three computers are involved: yours, mine, and the server. Let's say it takes 1 second for my computer to contact the server, and then another second for the server to contact you (and vice versa)
Say our planes stay on the collision course until the last second. 1 second before I see the collision on my FE, I jerk up on the stick. My FE, since it doesn't have up to the second info on where you are (remember, my FE is displaying 1 second old data), uses its smoothing code to predict that you continued flying straight at the same speed. Since I changed direction, my FE instantly displays that (no lag). However, the smoothing code assumes you went straight, and so on my FE you zoom under me. Collision avoided... or so it appears on my FE.
Now on your FE, you also see my plane 1 second from colliding with you. Say you take no evasive action and fly straight. In real life, I'm pulling up on my stick and avoiding the collision. However, that data will not reach your computer for another 2 seconds (1 to the server, 1 from the server to you). So your computer's smoothing code predicts that my plane flew straight. Your computer, based on this prediction, calculates a collision (as far as it knows I flew straight) and assigns damage... you take damage.
Now, your computer sends a "Brenjen hit Tarmac and died" message to the server. I didn't see a collision on my FE, so my computer simply sends a "Tarmac is climbing" message to the server... which relays that to you, and you see my plane climb away unharmed. The server sends me the "Brenjen hit you" message, along with the data that your plane has lost a wing and is in a death spiral. That data takes a second to reach me, so I see your plane fly unharmed for a second (since my smoothing code is still thinking you're flying straight and level), but a second later it loses its wing and begins to go down. Since I didn't see the collision on my FE, I take no damage.
Of course, the times are measured in tenths of a second, not full seconds, and these messages are sent multiple times per second. I used round numbers, but I hope you understand what's really going on.