"Saving" one's life was keeping your life your own and not allowing anyone, including God, to interfere with how you live it! Jesus said that to do so would actually cause you to "lose it." This is seen in two ways: as an unbeliever, not to give your life to Christ would mean losing your soul for eternity. Now that would be an eternal loss!!
The other way of "losing it" means that a believer can waste their time living for the pleasures of this world and gaining nothing of eternal reward. They would be saved, as Paul says in 1 Corinthians 3:15, "yet so as through fire." They would go to heaven because of the grace of God, which is true of ALL believers, but they would have nothing to lay at Jesus' feet. See Revelation 4:10, "...cast their crowns before the throne..."
Those people who give up their self-centered lives to come to Christ may seem to "lose" their lives according to the world around them. Think of how many people think that becoming a Christian means you won't have any fun! Little do they realize that losing the sinful "fun" now means you gain eternal pleasure in glory! Psalm 16:11, "You will show me the path of life; in Your presence is fullness of joy; at Your right hand are pleasures forevermore."
For the Christians who dedicate their lives for Christ this means "giving up" a life of mediocrity and gaining a life of absolute joy in serving the Lord! John 10:10 Jesus spoke of a life more abundant. That's when you follow Jesus and let Him have His way with you. There is no more exciting life than living your life for Christ!!