Why do Christians seek to convert everyone? This may touch a nerve, but why do Christians seek to convert everyone, especially people who have no interest in being converted? It always comes off like a drug dealer saying, "Try it, you'll like it - trust me. Would I lie to you?" If the point is to make the world a better place, wouldn't a less aggressive approach be more effective?
This answer presupposes that the claims of Christianity are true.  Since Christians believe this, and this site is meant to give a "Christian" view of things, that's the presupposition required to properly answer this.  I realize that many people don't believe the claims of Christianity, but the Truth of the claims isn't what's being asked or what's relevant here.  Why Christians act as they do is, and the presupposition is what drives us to act as we do.

This also is geared toward answering from the perspective of the strongly evangelistic among us. Not all denominations, and not all Christians are aggressively evangelistic.  From the question, I am assuming that you are looking for what drives that type, since that type is the ones that are "out to convert everyone".  The motivation of those that aren't aggressively evangelistic are going to be absent from my answer because it wouldn't apply to the question asked.  But for those of us who are, this is the reason:



That's simple. The idea of anyone in Hell terrifies us.  We believe that we've all fallen short of the standard God uses to judge us, and are all guilty, and therefore doomed to Hell. 

http://bible.cc/romans/3-23.htm


  John 14:6 New International Version 
  
  for all have sinned and fall short of the glory of God,


Since Christ claimed to be the only way to avoid that fate, naturally, we want everyone to turn to Christ and avoid that fate.


  John 14:6 New International Version
  
  Jesus answered, "I am the way and the truth and the life. No one comes
  to the Father except through me.


Many Christians feel uncomfortable with aggressive evangelism, stating that they're afraid to scare people away. Ray Comfort replied to this with "where are you going to scare them to, Hell #2?"

The non-believer may not agree that Christ is the only way, but belief isn't what determines truth, and if Christ is who He said He is, the only way to prevent someone from an eternity of torture is to tell them.  

Put another way, if someone were standing in the road, and you saw a vehicle rushing toward them, how would you react? By saying, "Please move out of the road!  Try it, you'll like it!" Or would you tell them of the danger, and if they ignored it, try to aggressively push them out of the way, and to safety?

Put simply, the point is not to make the world a better place. This world and our present lives are, if Christianity is true, only temporary. The point is to save those around us from eternal suffering.