Persecution is one of those words that we don't use much in our everyday language. It can be difficult to understand. Many times we think of biblical persecution such as with the apostles and what they endured. That is not the only type of persecution, however. Persecution can be when the majority, or close to, are against you for your belief, color, ethnicity, or the likes.
So, in this verse Jesus talks about being persecuted for righteousness' sake. In other words, people coming against you for trying to live a Godly life and do what is right in Gods eyes.
I look at it this way - Trevor Brazile is arguably one of the most talented cowboys around. I don't know this for a fact, but he probably faced many that did not like him simply because of his talent. We have all run into that at times. Just because you have some ability that someone else does not have, they decide they don't like you.
God knew this too. That's part of what Jesus says here; you are blessed when people turn against you because you desire to serve God and your life is committed to Him. This is not a command to go beat people with your bible. We can do more harm by preaching at someone than we can by the way we live and how we talk to them. This is telling you about godly living and it shows through your life. Not always in word, but in deed is how we accomplish this.
~ Chaplain Buddy Strachan