Some dental problem
First, it's great that you are flossing
If something breaks, chips, loosens or if the floss gets caught or frays when flossing, it indicates that something is wrong. It's better to floss and have this happen then to not floss and not be aware of some dental problem before it's too late and you have a bigger problem. See a dentist to find out what's wrong and what needs to be done to correct this.