So what does the Bible actually say? The verses which are quoted to support the idea that all non-Christians are always damned are John 14:6
Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me.” (NIV)and Acts 4:12
“Salvation is found in no one else, for there is no other name under heaven given to men by which we must be saved.” (NIV)For either to imply that all non-Christians are unsaved, we must assume that Jesus is unable or unwilling to work except in those who identify themselves as Christians; that He is kept out by any kind of ignorance on this topic, or that He has no power where there are no Christians. This is assumption is clearly unacceptable to a Christian when squarely looked at, but it rarely is so examined. These verses do say that Jesus's way is the right way and Mohammed's or Buddha's way the wrong way, but that does not imply that therefore Jesus is incapable of saving any Muslims or Buddhists without help from Christians. The Bible doesn't get that specific in terms of what happens to those who don't hear the message from us, but there are hints :
I have other sheep that are not of this sheep pen. I must bring them also. They too will listen to my voice, and there shall be one flock and one shepherd. (John 10:16, NIV)
The Bible is clear that those who have faith in Jesus are saved and those who reject Him are damned, but it does not say much about those who haven't heard His name at all or those whose idea of Him is grossly incorrect. In fact every example I can think of discussing people's eternal disposition divides everyone into categories based on accept or rejecting Him, and they picture that as happening at the Judgement not before death.
All this is not to say that there isn't value in spreading the Word; the Bible makes that clear, and we are unequivocally commanded to do it. It is only to say that failure to do so is not an automatic sentence of damnation.
I think there are several contributing factors making idea that non-Christians are all unsaved so common.
1) A Christian backlash against two unbiblical doctrines; that of Universalism that states everyone without exception is saved, and the warm, fuzzy, but anti-rational concept that all paths lead equally. The first (Universalism) assumes a God that takes you whether you want him or not; to borrow a phrase from Hank Hanegraaff, not the lover of your soul, but a supernatural rapist. The second (all paths are equal) makes as much sense as saying Aristotelean, Newtonian, and Einsteinean physics are all equally good for calculating an orbit. Both are extensively refuted in Scripture.
2) A meme. I'm not fond of the term because it's commonly used as a substitute for debating an idea on it's merits, but we addressed those already and "meme" applies well. The idea is simple and very motivating in terms of evangelism; those who have it pass it on to others efficiently.
3) Human desire to be in control. All religions are constantly fighting the human tendency to turn them into magic in order to put the religious practitioner in control. Religion is about describing supernatural forces, and also requesting their help; the decision and most of the initiative remain with the supernatural. Magic is about ways to effectively control the supernatural forces; do X and they will respond in Y way. The initiative moves to the human.
In evangelical churches the world tends to be viewed as divided into the saved and the unsaved, and they are divided by participation in what's very like a magic ritual; the recitation of the Sinner's Prayer. (There isn't actually any Sinner's Prayer in the Bible, but it's one of a whole class of prayers made up with reference to Biblical principles.) The distinction I'm making between magic and religion can get pretty fuzzy in the case that the supernatural side takes the initiative to make a promise, which God does in this case. Jesus does in fact say that he will save us if we believe:
I tell you the truth, whoever hears my word and believes him who sent me has eternal life and will not be condemned; he has crossed over from death to life. (John 5:24, NIV)We tend to make it a magic ritual rather than the true worship of God when we imagine it all depends on us; we reach the lost, we convince them to recite the prayer; it's like we saved them.