Two things are happening in the Sermon On The Mount. First, Jesus is letting everyone know that the standard is being raised (actually it was always this high) to a level that no human is going to be able to live on their own.

Second, He was introducing a way where we would all be able to live the life and do the ministry of love. Not killing anyone is good. But, if we live our life and do our ministry while hating them, our main purpose for living and ministering is not being fulfilled. The truth of the matter is, even this requirement isn’t high enough for the disciple.

Not murdering, not being angry, not speaking contemptuously and insultingly, and not calling someone a fool still will not get the job done.

First, we have to forgive. And not just once.
Luke 17:3-4 (NIV) So watch yourselves. If your brother sins, rebuke him, and if he repents, forgive him. If he sins against you seven times in a day, and seven times comes back to you and says, ‘I repent,’ forgive him.

Then, we have to love them.
Luke 6:35-36 (NIV) But love your enemies, do good to them, and lend to them without expecting to get anything back. Then your reward will be great, and you will be sons of the Most High, because He is kind to the ungrateful and wicked. Be merciful, just as your Father is merciful.