How can one offer Christ's sacrifice to God? I'd like to know what is the Catholic theology behind offering Jesus's suffering to the Father. 

I'm trying to understand this prayer (from St. Bridget's 12-Year Prayer) more deeply:


  Eternal Father, through Mary’s unblemished hands and the Divine Heart
  of Jesus, I offer to Thee the terrifying suffering of the Heart of
  Jesus on the Mount of Olives, and every drop of His Sweat of Blood, as
  atonement for my and all of humanity’s sins of the heart, as
  protection against such sins, and for the spreading of Divine and
  Brotherly Love.


I don't fully understand how can one offer somebody's else suffering to gain grace for oneself.

I have a theory but I'm not sure if it's correct. My theory or understanding is that none of us is capable of such a sacrifice which brings reconciliation with God. So offering it up to the Father it reminds him of this reconciliation. And here I think is the problem with my understanding. I don't think God the Father needs to be reminded of his suffering. That's why I'm asking what is the teaching...
Part of the answer can be found in Colossians 1:24.


  Now I rejoice in my sufferings for your sake, and in my flesh I complete what is lacking in Christ's afflictions for the sake of his body, that is, the Church.


The question a Christian must answer is, "What is lacking in Christ's afflictions?"  I think most would agree that nothing at all is lacking, so what does Paul mean?

From a Catholic worldview, when we are baptized into Christ, we are part of the body of Christ.  Being part of that body is not simply metaphorical but actual and real.  Saint Bridget has made Christ's sacrifice her own by uniting herself to that sacrifice in her devotions and selfless life, forming herself ever more into the likeness of Christ.

God the Father is well aware of Christ's sufferings, as his mother is and as we should be. We are called as Catholics to unite our sufferings to Christ's.

Bridget in her prayer is acknowledging Christ's sacrifice and joining herself to it, like Paul suffering for Christ's sake for the Gospel to our benefit. In this prayer, she points out the blood that the Lord sweat from his skin in the garden.

By combining this prayer with the Hail Mary and Our Father, Bridget shows this prayer to be contemplative prayer, like the rosary.  In contemplative prayer, she prays not just by repeating the verses, but rather by putting herself in the garden in the anticipation and fear that the Lord experienced. By imagining herself as Jesus and experiencing the fear and pain of what is coming, including the falling away by the few that should have been faithful, she joins in that suffering. Contemplative prayer is a powerful way to unite to Christ and experience his suffering.

In your question you asked:


  I don't fully understand how can one offer somebody's else suffering to gain grace for oneself.


Bridget is not offering just anybody's suffering up to the Lord, only the suffering of the one person whose suffering can and does merit grace.