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.
Since Christ claimed to be the only way to avoid that fate, naturally, we want everyone to turn to Christ and avoid that fate.
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.
It's because of The Great Commission given by Jesus Christ.
It's the duty of all Christians to share the Gospel which is Salvation through Jesus Christ and Him alone. Any Christian who does not share the Gospel are not completely following the command of Jesus Christ nor truly love Him.
First of all, some people may evangelize in ways that would be considered "aggressive", but many others are not. In fact, the situation often determines the appropriate approach, so the same person could approach the same person in completely different ways, depending on the situation.
When my mother saw me as a child wandering close to the street when no cars were coming, her voice was firm, but not aggressive. Then one day when we were walking and she saw a snake ten feet in front of me, her voice was quite different. Also, she has suggested very quietly that one more cookie may not be the best for me.
It should be noted that Christians do not merely believe that Christianity is something good that everyone would enjoy--like ice cream or a foot massage. A better illustration--and a biblical one at that--is that every person is infected with a deadly disease that impacts their bodies, their souls and their spirits. However, many people don't even know they have the disease. They experience the symptoms--spiritual emptiness, vanity, and others--but many do not even believe they have a spirit, so they don't know how to even self-diagnose.
As Christians, we have shared the same infection, but have received the antidote as well. It has transformed our lives entirely. We were also given an endless supply of the antidote to share with others. So, we share freely. Sometimes we may sound belligerent, but that is not from self-righteous, but from confusion as to why someone would not want to be healed.
We learn in time that some people simply do not want to know or admit that they have a disease. People even do this for physical diseases and won't go to a doctor. So, we have to learn to be tactful in how we share the antidote. Sometimes it is better for us to talk about how great it has worked for us. Perhaps that works well because people are more ready to hear they have a disease if they know there is a cure.
There is, indeed, a cure for the disease of sin and separation from God, and the cure is 100% effective.
I hope you can understand that even though you may find it annoying, the intent of evangelism typically comes from a very caring, loving and compassionate heart, albeit poorly expressed at times. Mom pleads with us to take nasty tasting medicine, because it heals and is good for us.
So, perhaps you can at least appreciate a heart that cares for you. If someone had a cure for a deadly disease and withheld it from you, you would have reason to hate that person. Yet, it someone believes you have a deadly disease and aggressively offers you the cure, at least they care. (This would be true even if it were not true that you had a deadly disease, because they believe that you do.)
So, I understand that many people do not really want to hear about Christianity. However, there are countless stories of people just like that who end up taking the antidote and receiving the joys that come with it.
Just to name a few accounts... The Unexpected Journey, The Persecutor, Betrayed, Once an Arafat Man. The list is literally millions of pages long...
A better question is, why wouldn't we?
We have the truth, and the truth is that you may have eternal life by believing in Jesus Christ. Pardon me if I say so myself, and I think I speak for all Christians when I say this, but eternal life is not a bad thing. In fact, if we didn't want to convert anyone, if we just kept to ourselves, it either means 1) we don't think we have the truth, or 2) we're selfish individuals who don't care about the rest of humanity. Neither is the case.
Now, I'll speak for myself. Of course not everyone wants to be converted. That's just the nature of fallen man. Jesus said (John 3:19-20), "And this is the condemnation, that light is come into the world, and men loved darkness rather than light, because their deeds were evil. For every one that doeth evil hateth the light, neither cometh to the light, lest his deeds should be reproved."
I know that's true. When I was in the world, I could have cared less about Jesus and anyone who preached to me about him.
David's view represents a Western, largely Protestant worldview. From an Eastern Orthodox perspective, it is because we can't help but share our loving God with you. But we don't do it by being pushy and trying to compel you to join our church. Instead we love you as God loves us. We don't believe we get people into the doors of the church, that is your choice and it is God who compels you. But we need to leave the doors open. The major difference is that in the East we focus on God's mercy, love, and grace, because God is love (the West tends to focus on God's wrath/anger). We also do not believe that God saves you from God (the Western view is that Jesus died to appease God's wrath in order to reconcile God to us). We believe that Jesus died to defeat man's enemies (sin, death, and the devil), and God's disposition towards all humans is loving, not one of wrath. We are the ones who need to reconcile with God (not vice versa), which is the state we were created to be in.
I liked David's answer and +1'd it. He talks about one working motive and he is absolutely right in that regard - because a Christian cares about the one they are witnessing to - that's the answer in the Penn Jillette perspective basically. I don't necessarily see a problem with those answers, and when I share with someone, it is out of a motive of love for that person, in part. But that is not the only reason, or even the primary reason that I get from the Bible. The primary reason I get is:
Because we are commanded to
The Bible commands this to us in Matthew 28:18-20 (NASB):
And frankly, that is enough. God deserves glory out of His creation. The only way He will get glory out of sinful, fallen mankind is through the transaction that took place at the cross. Through Christ declaring a sinner as righteous because His Son's punishment was sufficient for that sin and His resurrection was accomplished. And then the Holy Spirit taking up residence in that person's "heart". Salvation is about a gift to us, indeed, but it is about God receiving the glory due to His name. So "the Lamb that was slain may receive the reward of His suffering."
Let me ask you this. If you knew that a bomb was in a building and it was going to go off soon would you tell others? You are safe even if that bomb goes off, but those people are not shielded. If you don't warn people, then how will they know the danger they are in?
This situation is different of course, but the outcome is still the same.
I am a Jehovah's Witness and I go out every week (or I try to at least) to spread Gods message . It's not just that Jesus told us to "Therefore go and make disciples of all nations". We are generally concerned about people. We even print books and material at our own expense and give it away free.
We don't try to push our materials on people. If people are interested in bible knowledge we make sure they get it. If they don't want it or ask us to leave and never come back then we respect their decision. Even if people aren't interested or are downright nasty, at least you tried to save them.