How can one offer Christ's sacrifice to God?

I know that this is an old thread but I came across it and thought I would contribute. This is how I think of it.

The history of sacrifice is that a true sacrifice involves offering up only something of immense value to yourself. For example, in a time when life was hard and raising livestock required a lot of energy and resources, Abel offered his best animal, while it is implied that Cain's offering was not his best offering.

If you read the Gospel of Matthew, it is clear that the narrative is building up the idea in the reader's head that Jesus was the perfect sacrifice.

As Catholic Christians, we are called to a life in which we sacrifice the best we have to offer, in order to inherit eternal life. The best we have to offer collectively, as humans, is the life of Jesus, as the perfect human being, and he chose to be that sacrifice. So he offers himself to the father, and we offer him to the father also. The way that we offer him is through the Eucharist.

I did a little research and I found some answers.

Through his intercessory ministry in heaven and through the Mass, Jesus continues to offer himself to his Father as a living sacrifice, and he does so in what the Church specifically states is "an unbloody manner"—one that does not involve a new crucifixion.

"Do this in remembrance of me" can also be translated as "Offer this as my memorial sacrifice." The Greek term for "remembrance" is anamnesis.

So I think here is the answer. We offer his sacrifice at mass and in the prayers in remembrance of Him. And every time we remember we open ourselves up to the Lord (the Father). So it's this sacrifice which brings reconciliation with God the Father. I guess every sin we commit require reconciliation and offerings to God. And Christ's offerings(sacrifice) servers this purpose.

For this is that which was proclaimed by the Lord:

‘In every place and time let there be offered to me a clean sacrifice. For I am a great king,’ says the Lord, ‘and my name is wonderful among the gentiles’

[cf. Mal. 1:11]" (14:1–3).

More can be found here.

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.