What does the Bible say about giving your life away for Jesus? On a Christian radio station I constantly hear the phrase "to give your life away for Jesus" but they never say what that means.

I am very confused about this concept. Any clarification is helpful, thanks !
The question is a bit vague, and I would be curious as to what verses they were referring to (if any) so I could better answer the question, but I'll take a quick whack at it.

Things were perfect in the garden of Eden. Jesus was the ruler and there was no sin. When Adam and Eve chose to partake of the fruit of knowledge of good and evil, they brought sin into the world. Therefore, they lost direct connection to Jesus. 

When Jesus died on the cross He once again opened the gateway to God through Jesus (this is the significance of the veil tearing, but that is different story). 

God was in control when man was perfect, we gave up that control to ourselves. Jesus knew we couldn't do it on our own so He sent the Holy Spirit to instruct us. The Holy Spirit can't control what happens on the earth unless we, humans, allow it. Jesus will never come against free will. Satan doesn't have that power either, all he can do is trick us into believe wrong things. It is our choice and will always be our choice. 

When someone "gives up their life for Jesus", they are asking for Jesus to come and take control of their lives. They are giving up their right to accomplish things on their own strength. Once this surrender happens, Jesus can work in ways unimaginable. 

"I can do all things through Christ who strengthens me" - Phil 4:13


To taste that strength, you have to be willing to give up your free will and to give it to Jesus.