One of the many reasons Catholics feel so connected to Mary because of her humanity. Mary, although conceived without sin, lived a normal, human life. She was raised by parents who loved her, would eventually get married, give birth, and raise a child. Her child would even "run away" at one point, causing her to feel the panic any parent would feel.
But before Mary was raising the Son of God, she was a child just like us. When put into perspective that Mary was approximately thirteen years old when the angel, Gabriel, appeared to her and told her she was going to have a baby, Mary seems even more impressive. At the age of thirteen, she had probably had little to know experience with puberty and "being a woman". Just like my youngest sister, she was going through the hormone-craziness anyone in their early teens would. And she still said yes to God.
Mary, despite being so young and having no understanding about what she was saying yes to, said yes. She knew that she would be ostracized and unwanted, that she couldn't explain this to anyone without sounding crazy, but she loved and trusted God so much that she accepted whatever God's will was for her, even if that meant being a mom at such a young age. Just like us, Mary was a child of God and she knew that her good, good Father had a plan for her.
Which is why Mary is so easy to relate to as Catholics. We feel a kinship towards this young woman who gave up control to God so completely.