There are two ways of looking at this phrase.
Coming to grips
The first is to see that we are who we are and that we've become that—thus far. Sure we'll acknowledge those who God used to help form and fashion us. But beyond a certain point (in our lives, in our thinking), we neglect and throw off any notion of dependence upon another and take responsibility for our actions. This is what it means to "grow up". My dad always said that "the day you accept responsibility for your actions is the day you become a man". How true. And I can't tell you how many times I've looked back on my responsibility-less childhood and pined away for the freedom that came with the territory. I'm getting somewhat off topic here, but I will say that the only way to reconcile the two (freedom plus responsibility) is to walk before the Lord (see Micah 6:8 and 1 Corinthians 7:22)... I digress.
The attitude of needing to prove yourself (your position, your attitude, your mentality) to the world is one of the last vestiges of childhood. The fact that it is essentially a childish device—a tactic at attention-grabbing—means the rest of the big, bad, grown-up world usually—usually, but not always—is able to see right through it. With nothing to prove essentially means that you've come to grips with yourself and your place in the world. The outlook doesn't necessarily have to appeal to God because, let's face it, you're satisfied. On another note, one of my favorite quotes in the whole wide world is "Ah, but a man's reach should exceed his grasp, or what's a heaven for?" from the poet Robert Browning. What he's saying—as applied to the Christian's walk—is that we should never be satisfied with our station. Content, yes (see Philippians 4:11). But not complacent or apathetic. When you see a person walking around with nothing to prove yet content, you know that they've reached a level of adult sophistication where they live from a center that is a mixture of discipline, pragmatism and a hopeful outlook for the future. And hopefully gratitude to God as their Creator and sustainer.
Don't forget: "He giveth to all life, and breath, and all things;" (Acts 17:25).
One thing to watch out for with the person with nothing to prove is a too-easy willingness to inform the rest of the world of their station. If someone needs their position of satisfaction itself validated, I would wager to say that they haven't received the status of nothing to prove from God, so its basis is shaky at best. There may be storms ahead.
Grasping at straws
And this is where the second interpretation of the phrase comes in to play. This is why there's an ellipsis. Have you ever heard that old Christian-ism that states "if you were ever tried as a Christian, would there be enough evidence to convict you"? With reference to that phrase, the Judge in question must be God. And this is where things get serious. Before I go any further, God Himself asks us to "prove Him". Where the King James Version says "prove", it really means "test" or "try out": "Bring ye all the tithes into the storehouse, that there may be meat in mine house, and prove me now..." (Malachi 3:10 emphasis mine). What God is saying through Malachi here, is for us to overcome our doubts (in any and every area) about God and launch out on faith, expecting Him to be there when you put your foot down over the void. So, even though God is absolutely trustworthy, He's also willing to humble Himself and meet us where we're at. "Friend, go up higher" (Luke 14:10).
God has done His part. He sent Jesus. Who in turn lived the perfect life and gave it to us in the form of a re-created spirit with the auspices (The Holy Spirit, His grace) that enable us to renew our mind to His word, to develop "the mind of Christ" (2 Corinthians 2:16). Now it's up to us. God does expect us to "work out [our] own salvation..." (Philippians 2:12). To prove to God that He didn't waste His time in making us. In redeeming us. This is the state of play. This life—the life that we live—is our chance to prove to God that we have what it takes. And with nothing to prove that we're Christians, how does that reflect on the God who says we are? Christians, that is, i.e. like Christ.
"I can do all things through Christ who strengtheneth me." (Philippians 4:13)
And in front of a world that views with skepticism such highfalutin' concepts as Christianity purports, it behooves us to implement the above verse.
Prove it.