The meaning of this verse seems to be that the people would keep the law with more heartfelt earnestness, and consequently acceptably. There seems to be a reference to Numbers 8:7; Numbers 19:9, et seq.
Related Commentaries of Zechariah 13:1
Zechariah 13:1
1 In that day there shall be a fountain opened to the house of David and to the inhabitants of Jerusalem for sin and for uncleanness.a