Albert Barnes on Matthew 5:10

Persecuted. To persecute, means literally to pursue, follow after, as one does a flying enemy. Here it means to vex, or oppress one, on account of his religion. They persecute others who injure their names, reputation, property, or endanger or take their life, on account of their religious opinions.

For righteousness' sake. Because they are righteous, or are the friends of God. We are not to seek persecution. We are not to provoke it by strange sentiments or conduct, or by violating the laws of civil society, or by modes of speech that are unnecessarily offensive to others. But if, in the honest effort to be Christians, and to live the life of Christians, others persecute and revile us, we are to consider this as a blessing. It is all evidence that we are the children of God, and that he will defend us. All that live godly in Christ Jesus shall suffer persecution, 2Ti 3:12.

Their's is the kingdom of heaven. They have evidence that they are Christians, and shall be brought to heaven.

(f) "for righteousness's sake" 1Pet 3:13,14