First of all, realize that “the Christian way” is not a human invention. No Christian believes that they have come up with the best way to approach God all on their own. The way we follow was given to us in God’s word; it’s not our own. (The issue of why we accept the Bible as God’s word is another topic.) So we “believe” we are right not in way that we “believe” that red is a nicer color than blue; we “believe” in the same sense as we “believe” the Earth is round: Because we have received it from a trusted authority.
As to why we so much want others to also believe, it’s because we hate to see someone get hurt. For us to stay silent and keep our faith to ourselves would not mean “live and let live”; it would mean “live and let die”. Maybe this example will help:
Imagine humanity on one side of a great chasm, and God on the other. The people who realize that they are on the wrong side, that they belong on God’s side, are trying desperately to get across. Some try to back up and get a running jump; they fall short – literally. Others try to swing a grappling hook across, but there’s nothing for it to catch on. Some try to climb down into the chasm and up the other side. But the walls are so sheer, they just slide screaming into a bottomless pit. Some try to go around the end of the chasm, only to learn that it forms a circle, enclosing humanity, separate from God. Some, whom we greatly admire, start building bridges across the chasm. They get halfway across, and run out of materials or time. Or it looks like the gap is bridged, but there’s not enough strength to support their weight as they try to walk across. And each of these has others following in their footsteps, trying and re-trying all the ways that don’t work.
And of course, there are always those who just don’t care, or pretend not to. They don’t even attempt to get to God’s side. In fact, they act as if God doesn’t exist, or if He does, that’s of no consequence to them. They make themselves comfortable on humanity’s side of the chasm. What they don’t realize is that when we reach the end point of God’s plan for His world, their side will cease to exist as God remakes the world and eliminates anything separate from Himself.
Meanwhile, the Christians have found God’s directions to the bridge that He built, the only one that gets humans safely across to His side. It takes just a little effort to get to it, mostly the effort of abandoning all the other methods of trying to cross. But it’s wide and strong and sturdy and dependable; it works. We’re standing on this wonderful bridge, that we did nothing to build, and having to watch the futile efforts of those we care for.
Do you wonder that we almost frantically scream out “Please, come over here. Don’t keep trying so hard at something so doomed. We’ve found the answer. Please, oh please, go across with us”? To be ignored and have to watch others be lost is heartbreaking.