While one cannot speak for their motivations or methodology, Muslim terrorists use the Qur’an the same way that Christian extremists such as the Ku Klux Klan and Aryan Nations or Jewish extremists such as Meir Kahane and Baruch Goldstein in Israel use the Bible: by taking phrases out of context and developing interpretations that serve their agenda.
They also ignore principles of the interpretation of texts followed by legitimate scholars of religion, above all the principle that a text must be understood with reference to the time, place, and situation in which it was given. The Qur’an, like other seminal religious texts, has a dual nature: one that is specific (particular or transitional) to the occasion, time, and place, and another that is universal and permanent, dealing with principles that apply for all times and places. The specific cannot be made to apply universally, while the universal always informs the specific. Ignoring this principle leads to arbitrary interpretations tailored to fit political agendas.
Most of the terrorism committed by people claiming Islam as their motivation is justified by a methodology that bypasses the bulk of classical scholarship. Various legal issues that pertained to the majority of the Muslim community were often left to the discretion and judgement of qualified scholars. ISIS and other similar groups, however, discount the role filled by traditional scholars. They promote themselves as “scholars” and then produce rulings far removed from what Muslims traditionally would find normative, acceptable, or humane.