During WWI, it seems like pilots would just get in their respective planes to shoot other pilots down that got in their planes. What was the point of pilots shooting each other down? Why not just stay on the ground and everyone lives?

The western front was largely an artillery war. Though infantry is ultimately the only arm capable to occupying and holding ground, artillery in WWI was the decisive factor.

When bombardments failed or an attack lost it's barrage, or when the infantry outrun their own artillery, the results were invariably a bloody slaughter. On the other hand, when the infantry and the artillery co-operated, the objectives would almost invariably be achieved without excessive losses being incurred.

One of the critical factors in developing an effective artillery arm was the ability to spot - to spot fall of shot, to spot enemy fortifications, to spot enemy batteries, to spot the location of your own infantry - and to report these spots back so the artillery could adjust their accordingly and avoid an attack degenerating into a bloody repluse.

Now aircraft (and observation balloons) are a pretty good way to spot these things. They can take photos and bring the film back for development and analysis or they can signal the ground using a primitive radio transmitter (radios at the time were too bulky for a proper transponder radio so they tended to only transmit - even so, tapping out a morse code message while flying a plane made of wood and canvas was no mean feat).

So there's an obvious advantage in being able to do this while denying your enemy the chance to do the same. So we start to get aircraft trying to shoot each other down. Soon they are armed with machine guns to defend themselves or attack the enemy. Pretty quickly we get the evolution of the dedicated fighter aircraft, whose only purpose is to shoot down the enemy.

This leads to aerial combat as we recognise it, with fighters contesting for air superiority so that they can deny the enemy the freedom of the skies.