God wants us to humble ourselves under His hand. Our human nature tends to rebel against this. We want to be in control, but let's face it, we're not. We are dependent upon God for everything: our lives, our talents, our treasures, even our very breath. If God stopped sustaining us for even a moment, we would just disappear. To be humble is to recognize this fact, to understand and accept our position in relationship to God. It is not to cut ourselves down but simply to acknowledge that God is God and we are not and that we belong totally to Him and are completely reliant upon Him.
What's more, being under God's mighty hand definitely isn't a bad thing. God's hand is mighty, in the Greek krataios, strong, powerful. But God's hand is not mighty in such a way as to crush us but rather to protect us. We have an all-powerful Guardian, Who, if we humble ourselves and snuggle under His sheltering hand, will care for us in every way.
God will do even more than that. He will exalt us, raise us up, when the time is right. He will raise us up out of our miseries. He will, one day, in due time, raise us up right out of this world and into Heaven where we will see Him face to face.
But we must place ourselves humbly in His care first. God is a gentleman. He will never force Himself upon us. He offers us grace upon grace, but He requires us to respond freely to that grace day by day, hour by hour, minute by minute all the way to the very end of our lives on this earth. Then He will raise us up to Himself at just the right time.
The second verse here is not really a separate sentence in the Greek. The word translated as the imperative “cast” is really an aorist participle “having cast” or in Greek epiripsantes. This word starts a clause that modifies “humble yourselves.” So casting your anxiety on God is part of being humble. When we think about that, it makes sense. Anxiety is merimna in Greek, and it literally means something divided. Anxiety and worry divides us, breaks us into pieces, fractures our minds and hearts and sometimes even our bodies. It tears us apart. It can also be a form of pride. When we let anxiety overtake us and break us down, we are really trying to take control over things we can't change (and fussing because we can't change them) rather than trusting in God and allowing Him to work things out for us in His caring way. When we let go of anxiety, we humble ourselves before God and recognize that His is indeed in charge and will do exactly what is best for us.
Why? Because He cares for us. The Greek verb here is melō. God is concerned for our welfare, concerned in a deeper way than we can even imagine. He takes a personal, loving interest in each one of us. He wants us to be in Heaven with Him forever, and He gives us all the grace we need to get there. We, in turn, must humble ourselves before Him, casting our anxiety on Him and accepting this grace that God may exalt us into His presence and into His arms for all eternity.