That is an excellent question, and one more people should ask, including those who believe they need or ought to convert the world.
I believe you probably understand at least part of the answer already, and could give it yourself. There is a little bit of the evangelist in most of us. If we believe something strongly, there is an urge to convince others that have not yet discovered this truth, whatever it is, that means so much to us. This kind of evangelism is pretty much based on good will and a desire to share what we believe is true and good.
A more ardent and not always so innocent form of evangelism comes from a belief that one is mandated by their God or their religion to convert others, to "save their souls," from something. Often this kind of evangelism is mixed with a militant fervor, with either an implicit or explicit conviction that one is part of a Holly mission or plan that depends on one's doing their part. So long as this kind of militant evangelism is not combined with a belief that one is justified in using force, though it may be extremely annoying, it is not dangerous.
Finally, there is the rare form of religious conviction that one is mandated by God (or Allah) to convert others, or at least bring them under the control of, "God's law," and that they are justified in using force, or terrorism, or outright deception (if the opportunity for force and terrorism has not yet arrived) to fullfill their mission. This form of "evenagelism" has cropped up in many religions, but always as an anomoly and perversion of the main body of the religion. In the Muslim religion, it has historically been the dominate form, and in terms of raw numbers, it is the dominate form today.
Hank