Jesus said I am the Way, how does the Catholic Church best explain this passage?

I find your question a little hard to understand. I think you're saying that you think the Catholic Church must have difficulty with these passages, presumably because of the apparent belief that the Church functions as a mediator. The stereotype in certain Protestant circles is that the Catholic Church displaces Christ as the mediator of salvation. This is not correct.

Let's start with paragraph 74 of the Catechism of the Catholic Church, a lengthy summary of the Church's teachings:

God "desires all men to be saved and to come to the knowledge of the truth" (1 Tim 2.4): that is, of Christ Jesus (John 14.6). Christ must be proclaimed to all nations and individuals, so that this revelation may reach to the ends of the earth

So straight away, you can see it clearly stated that Jesus is the sole truth, and the sole means of salvation. This paragraph cites one of your verses, and the verse immediately before the other.

Elsewhere, we have paragraph 1698:

The first and last point of reference of this catechesis will always be Jesus Christ himself, who is "the way, and the truth, and the life."

Or paragraph 2466:

In Jesus Christ, the whole of God's truth has been made manifest. "Full of grace and truth," he came as the "light of the world," he is the Truth.

Or paragraph 2614:

Faith in the Son introduces the disciples into the knowledge of the Father, because Jesus is "the way, and the truth, and the life."

I think these clearly indicate the unique mediatorial role of Jesus.

With regard to the 1 Timothy passage, the key part of the Catechism is paragraph 618:

The cross is the unique sacrifice of Christ, the "one mediator between God and men". (1 Tim 2.5) But because in his incarnate divine person he has in some way united himself to every man, "the possibility of being made partners, in a way known to God, in the paschal mystery" is offered to all men. He calls his disciples to "take up [their] cross and follow (him)", for "Christ also suffered for (us), leaving (us) an example so that (we) should follow in his steps." In fact Jesus desires to associate with his redeeming sacrifice those who were to be its first beneficiaries. This is achieved supremely in the case of his mother, who was associated more intimately than any other person in the mystery of his redemptive suffering. Apart from the cross there is no other ladder by which we may get to heaven.

This is a wonderful explanation of Catholic ecclesiology. Christians have a share in Christ's saving action, uniting their sacrifices with the one perfect sacrifice of the Cross.

It is not for a second about diminishing the unique place of Christ's sacrifice, or his unique mediating role.

Similarly, we can see this in 1544f.

Everything that the priesthood of the Old Covenant prefigured finds its fulfillment in Christ Jesus, the "one mediator between God and men." (1 Tim 2.5) The Christian tradition considers Melchizedek, "priest of God Most High," as a prefiguration of the priesthood of Christ, the unique "high priest after the order of Melchizedek"; "holy, blameless, unstained," "by a single offering he has perfected for all time those who are sanctified," that is, by the unique sacrifice of the cross.

The redemptive sacrifice of Christ is unique, accomplished once for all; yet it is made present in the Eucharistic sacrifice of the Church. the same is true of the one priesthood of Christ; it is made present through the ministerial priesthood without diminishing the uniqueness of Christ's priesthood: "Only Christ is the true priest, the others being only his ministers."

The priesthood of the Church in this present age is precisely a participation in the one priesthood of Christ. The sacrifice of the Mass is precisely a participation in Christ's sacrifice on Calvary.

As you say, I think a correct interpretation of this is important for ecumenism. It is vital for the correct Catholic teaching to be understood by the Catholic faithful and by other churches, rather than the popular and misleading caricature that perverts the Gospel and sets Christians against each other.