Why do Martyrs exist in History?

Religion is not the sole reason for martyrdom. A martyr is someone who is killed because of it's belief, any belief, not just religious belief.

Actually God is well-able to give both proof and experience to people. Your personal lack of experience does not negate the experience of the person who has experienced. 

Because they choose to.

Some Christian martyrs didn't just believe; they actually witnessed Christ and his miracles. And some others received their information about Jesus second-hand from several  people whom they could trust their life on. And others witnessed miracles from the Apostles which the Apostles attributed the miracles to Jesus Christ. 

After thousands of years after the existence of Jesus (the man) it becomes less of a reasonable certainty and more of faith. Still, (and though it's easier to say than it is to actually do), I would fight until the death against any Marxist or atheist cause to stifle my faith, or the expression of my religion. 

Yes, you can believe in something so much that you would die for it.  Where you are wrong is that the belief has no proof.  

You unbelievers keep saying no proof. You just don't get it, we have the proof we need, God lets us see it, but not you because of your stubborn hearts, this is on you because you won't soften your hearts and let him in.

I guess the martyrs would beg to differ with your "no proof" statement.

If you don't stand for something, you'll fall for anything.

Its called faith. And the rewards are out of this world.

Matthew 16:25 KJV — For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it.

That goes to the nature of faith, Heb. 11:1, 6.  If the martyrs who lived during Christ's ministry did not believe Him, they surely would not have been willing to die for Him.

Because people supposedly died for a cause.