In Jn. 8:34 Jesus talks about some form of slavery that eventually would affect man's eternal fate. And this, after all, is what matters. As they say, in Philosophy, finis est super omnia (the attainment of the goal is above all most important). So Jesus states "everyone who commits sin is a slave." Since that is what conditions our eternal destiny, it must then be most important.
Study past and present history, look at our world today or in past ages, see how the havoc of sin has cost us so much misery and tragedy. How? There is pride, disobedient and insubordinate. Beginning with the rebel angels who were plunged into hell of everlasting fire and unhappiness (cf. Rev. 12:7-12). Jesus is the Way, the Truth and the Life (Jn. 14:6), hence the way to freedom from slavery is listening to His words and living according to His ways of truth and love.
In the Bible you can see how this slavery works. You can be a slave to your own pride, refusing to listen to God's inspirations and the legitimate authorities placed above you. "Pride goes before destruction, a haughty spirit before a fall (Prov. 16:18). Even St. John of the Cross, the Carmelite mystic said: "God desires the least degree of obedience and submissiveness more than all those services you think of rendering him."
You can be enslaved to power and material things. Jesus had said: "It is easier for a camel to pass through the eye of a needle than for a rich man to enter the kingdom of heaven" (see Mt. 19:23). How many acts of violence have been perpetrated for power and greed. Of course, when the apostles were disturbed by such statements of Jesus, they reacted saying "Who then can be saved?" Jesus must have looked at them with pity, saying: "For man this is impossible, but for God all things are possible" (see same paragraph above). Sexual excesses and sins of the flesh, as seen through media in our day, can also make people enslaved or addicted. The Church presents to us prayer and the sacraments as means of conversion through God's grace and mercy.