How did Mary become the Mother of the Church?
The Father chose Mary from among all women to be the mother, according to human nature, of His Divine Son. As she is Mother of Christ in the natural order, she is also the Mother of His Mystical Body, the Church, of which He is the Head in the order of grace.
In the Book of Hebrews 2:9-13, the author makes it clear that Jesus is our brother:
[W]e see Jesus, who for a little while was made lower than the angels, crowned with glory and honor because of the suffering of death, so that by the grace of God he might taste death for every one.
For it was fitting that he, for whom and by whom all things exist, in bringing many sons to glory, should make the pioneer of their salvation perfect through suffering. For he who sanctifies and those who are sanctified have all one origin. That is why he is not ashamed to call them brethren, saying,
“I will proclaim thy name to my brethren, in the midst of the congregation I will praise thee.”
And again, “I will put my trust in him.”
And again, “Here am I, and the children God has given me.” (emphasis added)
Since Jesus is our brother, and Mary is His mother, it follows that Mary is our mother as well.
Finally, when Jesus was on the cross, we see the tender moment when He gave Mary to the Apostle John. In the Gospel of John 19:26-27, we read:
When Jesus saw his mother, and the disciple whom he loved standing near, he said to his mother, “Woman, behold, your son!” Then he said to the disciple, “Behold, your mother!” And from that hour the disciple took her to his own home.
Thus, the Tradition of the Church, coming from the Apostles, teaches us to understand that the Lord was entrusting all of His disciples to Mary, in the person of St. John.