How do islamic terrorists justify their violent actions

Best  They believe they have a "just cause."

If they pursue their actions in the name of Islam, terrorists will justify their actions as a defense of the Islamic faith or of territory rightfully belonging to Islamic believers.  They interpret the Qur'an as being in favor of their "jihad" or holy struggle.

Terrorists may speak out against the evils of Western culture and attack symbolic targets of what they see as the heart of evil in Western culture.  But their terrorist actions themselves don't "demonstrate the evils of Western culture;" their actions are a response against what they would see as Western culture and its evils.

There are some Islamic groups (such as ISIS) that are looking for territorial domination over certain areas (such as the Islamic State in Iraq and Syria -- aka "ISIS").  But Islamic jihadists are typically not trying to achieve a global empire, but are aiming at specific goals in a specific region where they are operating.