Final answer:
The Florentines are hard on Farinata's family because they see themselves as heirs to the Ancient Roman Republic and perceive them as a threat to their freedom.
Step-by-step explanation:
Dante explains that the Florentines are hard on Farinata's family because they see themselves as the heirs to the Ancient Roman Republic and are deeply committed to the cause of freedom and liberty. The Florentines interpret their military victories and their status as the 'New Rome' as signs of God's favor and protection. This leads them to take strict measures against anyone they perceive as a threat to their freedom, including Farinata's family.