This paragraph describes the difference between the instant sanctification that happens when we receive eternal life through faith in Jesus, versus the gradual and progressive sanctification that happens as the Holy Spirit continues to transform us every day.
When we become His followers we are given new hearts, and we become slaves to Christ instead of to sin. But that doesn't mean that our struggle with sin is over, or even easy. As long as we remain in our earthly bodies we will continue to struggle with sin, but it is a battle worth fighting, and it will get better.
I heard once that as we mature in our Christian faith we will begin to sin less and less. However, our sin will bother us more as we continue to more fully understand the depth of our depravity and the utter holiness of God. So it might seem like we're not making any progress even though we are!
The key is to continue to fight against that sin nature in His strength. After all, as the scripture above says, Jesus Christ is the answer.