If you don't mourn for your sins, you'd mourn because of your sins. 

The remnant of Israel has gone ahead and intermarried from neighbouring countries,...

If you don't mourn for your sins, you'd mourn because of your sins. 

The remnant of Israel has gone ahead and intermarried from neighbouring countries, of which they were prohibited from getting married to. Ezra saw this and was grieved. 

The people of Israel had just seen the hands of God against them because of their disobedience and idolatry practices and this would be a setback for them. 

They all gathered at the house of God praying, confessing and weeping bitterly before the Lord for their sins pleading for His mercy. 

  Shechaniah, the son of Jehiel spoke to Ezra telling him, “we have been unfaithful to our God by marrying foreign women from the people around us… Let us make a covenant before our God to send away all these women and their children in accordance with the counsel of my Lord and of those who fear the commandments of our God.” Ezra 10:1-3 (paraphrased) 

  These people knew their errors, they were apologetic about it and in their sorrows they made adjustments. 

They took steps to making sure they were right again with God, that's Godly sorrow. 

Sorrow that makes us genuinely apologetic for our wrongs that we do things to make sure we are right with God. 

Apostle Paul in 2 Corinthians 7: 8-11 explains this to us as he was glad that His letter to the Corinthian church prompted a kind of sorrow in the people which led to their salvation, that in their sorrows they repented of their sins.

 As Christians, we should be able to sorrow over our wrongs, letting the truth of God's word heal and mend us, leading us back to true repentance.

 So instead of just saying, God I apologize for that lie I told every time seek for ways to stop situations that would make you lie, get conscious of your word. Seek truths from God's word that'd help you emerge victorious, and in your little efforts, the Holy Spirit would step in and heal you completely.