What seems to have been the justification for the extremes of violence condoned by the crusaders in the fall of Jerusalem?
I would say that religious fervor and a corresponding hatred of the Muslims who held Jerusalem is what led the Crusaders to justify the massacre of the people of Jerusalem when it was conquered in the First Crusade.
If you look at the eyewitnesstohistory.com link and you look at the statement by the Crusader, you will see how much religious thinking (and religious bigotry) played into the thought of that particular participant in the Crusade. This witness refers to the warriors as "pilgrims" as in the following quote:
Entering the city, our pilgrims pursued and killed the Saracens up to the temple of Solomon. There the Saracens assembled and resisted fiercely all day, so that the whole temple flowed with their blood.
He also talks about how the Crusaders were animated by a desire to repay their debt to Christ:
Afterwards our men went rejoicing and weeping for joy to adore the sepulchre of our Saviour Jesus and there discharged their debt to Him. . . .
The idea, then, was that the soldiers' Christian duty was to kill the non-Christians who had been holding the sacred city. As the eNotes link says:
It was God's will, the Crusaders believed, that they cleanse the holy city of unbelievers. When their thirst for blood was satisfied, they gathered in the Church of the Holy Sepulchre to pray.