Question: What are broken views (破见)? Why are they more gravely wrong to have than broken precepts (破戒)?
Answer: To have broken views is to have Right Views (正见) destroyed. Views destroyed might include the fundamental understanding (and thus right intention) for receiving (受戒) and upholding precepts (持戒). Without Right Views, there will be the tendency to have evil views (邪见).
As an example of an evil view, some might uphold the precepts just for fault-finding others; not for correcting oneself at all. (While it is alright to point out faults appropriately, there should be priority on correcting oneself first. Those who focus on fault-finding others are usually blind to their own faults.)
Those with Right Views, who have broken precepts will later repent for their unmindful lapses. However, those with broken views, thus lacking Right Views, and even having evil views, will neither understand nor recognise their breaking of precepts, thus continue being wrong. To prevent having breaking of views, there should be continual learning for bettering practice.