It means just that, offer it up to Jesus, in union with his sufferings, to the Father as he offers Himself. We offer everything–our joys, sorrows, sufferings, sacrifices, blessings, in a conscious way through words or silent intention, to the Father, with the Son, through the Holy Spirit. There is a formal prayer by which we dedicate the entire day, called the Morning Offering, or we make an intentional act whenever we experience a sorrow or a blessing. We give back to the Father whatever we have received, in gratitude, praise, thanksgiving, and petition for his blessings. In the words of the song, cast all your cares on Him, for He cares for you.