Think about it this way: God has promised to provide our necessities if we seek Him first (Matt. 6:33). God doesn't break His promises! Thus, if we really believe God, then we will not worry about our needs being met. And, let's be candid, if our needs are being met, what else is there to worry about? Death? That should not be a worry for a faithful Christian. We've already seen Paul declare in this epistle that it wasn't a worry for him (cf. 1:21).
Our prayers need to be full of thanksgiving as well as making our requests to God. Having a solid prayer life will help us to have the peace that comes from God. It is an inner peace and contentment that goes beyond human understanding, at least from the world's perspective. The world doesn't understand the joy and security of being in Christ. Even when we're struggling physically, we can have this inner peace knowing that God loves us and will take care of us. The confidence of knowing you are saved brings an inner peace that no one who is worldly-minded can fathom.
Paul teaches that this peace will guard our attitudes. God will make sure that we are not troubled if we are willing to give ourselves to Him in prayer and obedience and confidently trust Him.
We will continue studying Philippians 4 tomorrow.