What does it mean to "cast your cares on The Lord"? Well, it truly means to throw them on Him & trust that He's going to handle your situation. That being said, we should not throw them on Him with the attitude of, “You better or else...” We should throw cares, our worries, our burdens on Him with the trust that He can get us through the storm, the situation, the weight, etc. Here are some bible verses to read over yourself and memories.
“Cast all your anxiety on him because he cares for you.” (1 Peter 5:7)
“Do not be anxious about anything, but in every situation, by prayer and petition, with thanksgiving, present your requests to God. And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus.” (Philippians 4:6-7)
“Come to me, all you who are weary and burdened, and I will give you rest. Take my yoke upon you and learn from me, for I am gentle and humble in heart, and you will find rest for your souls. For my yoke is easy and my burden is light.” (Matthew 11:28-30)