The Koran originally started off as a version of the Bible and Torah. Mohamed identified himself as the final prophet, a messenger to the Arabs.
After ten years and only a few followers but getting kicked out of Mecca for saying "no idols" in a town that lived off religious tourism, he went to Medina. When he said he was the messiah to the Jews there, they shunned him but didn't hurt him. Twice rejected, he declared war on the world - death to all who will not submit to Islam.
He said Allah gave his followers permission to raid religious caravans to Mecca, as long as he got a cut. They had permission and indeed a religious mandate to terrorize a few with violence to scare the rest into converting to Islam. Hence beheading and showing off the heads, not just killing them. Hence religious verses to cut off fingers and a single hand, not merely murder.
In the next years, Mohamed became wealthy, took sex slaves, legitimized raping sex slaves, said it was a death penalty to leave Islam even if you converted in order not to be killed.
His verses include stating the only way a man can be guaranteed to go to heaven is to die for the spread of Islam. So many men die spreading Islam to guarantee going to heaven. What makes Islam unique is the mandate in the Koran to go to war to spread the faith, instead of merely going to war as followers and saying victory is because your God backed you.
The greatest good in Islam is spreading Islam. If you murder 10 to scare 1000 into converting, the murders are considered justified. If you maim 5 to scare 50 women into covering up per the Koran's mandates, Muslim or not, it is considered moral.
Islam says it is moral to lie to spread the religion, such as conversion, and to lie to protect the image of Islam as perfect.
Islam is also unique for the religious text mandating creation of a government that enforces the religion on the population. Shariah says if you pay extra taxes and submit to systemic humiliations (beatings when you pay jizya tax, cannot be given the same greeting as Muslims, distinctive clothes so others know to abuse you, half the worth of others in court or blood price, not allowed to own weapons for self defense), you can be non-Muslim. End result - 100 ways every day you're given reasons to convert while the jizya tax goes to fund jihad.
TLDR - Muslim terrorists are acting per the words and example of Mohamed, as their religion commands.