In his sermon "A Cure for Care," Charles Spurgeon delves into the biblical teaching of casting all anxieties on God, as referenced in 1 Peter 5:7. Spurgeon emphasizes that believers should not be burdened by worries and cares because God is both willing and able to carry these burdens for them. He encourages Christians to trust fully in God's providence and love, reminding them that worry is not only unnecessary but also a sign of lack of faith. The sermon provides practical advice on how to cultivate a worry-free life through prayer, trust, and the assurance of God's constant care and protection.