The breastplate protected the part of the body with the vital organs. Just like in practical warfare, it is vital in our spiritual battle.
Righteousness essentially means obedience to God's law. It is not by our own works, it is the righteousness of Christ imputed to us. We love the law of God and we therefore obey it.
Eph.6.14Stand therefore, having your loins girt about with truth, and having on the breastplate of righteousness;