Jesus just got through saying that you cannot be devoted to two masters. Inevitably you end up preferring one over the other. Specifically, you can't serve God and material possessions. You will either end up putting all your hope and security in the Giver or in the Things Given.
"For this reason, do not be anxious for your life." In other words if you want to avoid the trap of putting your hope and security in Things, if you want to stay focused on who your true master is, then here's the secret: Don't be anxious. Yep, that's right. All that anxiety about your material needs has to go. There. All better now?
But we need a little more help than that, something reassuring and concrete. So look at the birds. No, don't look at your online bank statement. Open the blinds or curtains or whatever and look at the birds outside. They neither sow nor reap nor gather into barns. Now at this point you have to understand that the people in Jesus' day lived in an agricultural community. Their form of anxiety involved sowing seed and anxiously waiting for the rains to come. Then when the rains came and the crops grew, they anxiously hoped that frost or locusts didn't destroy the crops before harvest time. Then after they reaped the harvest they stored the extra in barns as a safeguard in case next year's crop should fail. It was a continual cycle of anxiety at every stage of sowing, reaping and gathering into barns.
Our equivalent would be getting anxious over landing a job, then being anxious about whether the paycheck would be enough to cover the bills, then if there's some money left over putting it into a savings account as a safeguard against next month's expenses. So Jesus might very well say to us, "Look at the birds of the air. They neither land jobs, bring home paychecks, nor set up savings accounts, yet your heavenly Father feeds them. Are you not worth much more than they?"
I think the "worth much more than they" part is the most comforting. Our security can be anchored in knowing that God values our lives more than all the other creatures of the earth--and we see plainly how well he cares for them. I don't think we really believe that God takes care of us because we are worth so much to him. We think it's because we're trying hard to be good little girls and boys who make our beds and wash up and say "please" and look sorry when we are scolded. We act like we're in some 19th century Charles Dickens orphanage where at any moment we could be turned out into the icy streets if we violate one of God's petty headmaster rules. The truth is we're privileged sons and daughters in our father's own home, and he gives us everything we need in abundance even though we daily break all the major rules of house.
Since we're already in that safe and secure place, you can see how being anxious isn't going to add a single cubit to our lifespan as Jesus says. It's like this. If you're anxious every day, your Father will love you and provide for you. But if you're not anxious at all, your Father will love you and provide for you. How does adding anxiety to the mix help you to come out ahead? It doesn't. So lose the anxiety and rest in the provision of the Giver, then you'll see clearly who your true Master is.