"'And you will know that I have sent you this admonition so that my covenant with Levi may continue,' says the Lord Almighty. 'My covenant was with him, a covenant of life and peace, and I gave them to him; this called for reverence and he revered me and stood in awe of my name.'" NIV translation
In Genesis 49, when Jacob was blessing his sons, he called his son Levi a murderer. Yet, in Numbers chapter three, God chose the Levites (the tribe of Levi) to be set aside to serve Him. They were chosen to serve and to teach God's people His law. To do so required that they have proper respect for God and His Name.
"'True instruction was in his mouth and nothing false was found on his lips. He walked with me in peace and uprightness, and turned many from sin. For the lips of a priest ought to preserve knowledge, and from his mouth men should seek instruction - because he is the messenger of the Lord Almighty.'" NIV translation
God is telling these errant priests that He knows their father (Levi) and God is reminding them of how Levi acted towards Him. He reminds them that nothing but the truth (which is the Word of God) was found on his lips. Then God reminds them that the people should come to them for instruction because they are the appointed messengers from the Lord Almighty.
Once again, we may say that this is the Old Testament and that it doesn't apply to us because we aren't priests but Jesus said that all believers are a royal priesthood. Therefore, like these priests, nothing but the truth should pass from our lips and the lost (those that are not believers/priests) should be able to come to us for instruction.
"'But you have turned from the way and by your teaching have caused many to stumble; you have violated the covenant with Levi,' says the Lord Almighty. 'So I have caused you to be despised and humiliated before all the people, because you have not followed my ways but have shown partiality in matters of the law.'" NIV translation
Now, God has shown how they have turned their back on Him and so are being punished. In short, they broke their faith and instructions from God and chose to please men instead. As God is just, they are now disgraced in the sight of those very same people that they put before God. There is a very good lesson for us today as well in that, if we put others above what God tells us to do, we can expect to be brought low in front of them as God corrects His children/priests.