There is more to the story. This is the way that I understand it.
Cindy McCain was visiting an orphanage started by Mother Theresa. They were providing medical care to some of the children there.
Two little babies were very sick. They both had medical problems. Cindy was able to bring them to the United States for further medical treatment. I believe that Bridget had a cleft lip/palate (or at least the palate). So, I believe they brought the girls to the US very similar to what happened with Baby Alex that we fostered.
I think that Cindy fell in love with that little girl and then wanted to adopt her. I don't know much about the actual adoption. (like how it actually, legally happened)
Here's a reference:
This is from an online story in Harper's Bazaar
Cindy McCain: Myth vs. Reality - Bazaar.com
That wrath was reserved for Bangladesh's minister of health and human welfare, who in 1991 tried to stop her from leaving his country with two sick, abandoned baby girls, one of whom is now the McCains' 15-year-old adopted daughter, Bridget. In Bangladesh for a month with the American Voluntary Medical Team (a nonprofit founded and funded by Cindy), she arrived at Mother Teresa's orphanage in Dhaka to find "160 baby girls who'd been dumped. And I just came upon this one baby — though I absolutely believe Bridget picked me — with a cleft palate so severe that, if it wasn't fixed, she'd die because the nuns couldn't feed her. Each time we returned, I'd hold her, play, marvel even then at her tenacious strength. Finally I wound up getting medical visas for her and another baby with a heart condition — not thinking about adoption — just getting them help."
As she was leaving for the airport, Cindy got a message from the minister of health, "wanting to see me and the babies. I went, and there were all these men talking to each other while I sat with two infants, not understanding a word. Finally one said, 'We can do surgery on this child.' But they had no intention of helping her, and I had to make a plane, so I just slammed my fist on the table, shouting: 'Then do it! What are you waiting for?' They were so stunned, he immediately signed the papers. I don't know where I got the nerve. I could've been arrested."
By the time she landed in the United States, Cindy "realized I couldn't give up this child" and called her husband, telling him about her charges and asking him to meet her at the airport. "When I disembarked carrying Bridget, John said, under his breath, 'Where's she going?' I said, 'To our house.' He laughed. 'I thought so.'" She chuckles. "I brought home a baby without telling him, and he not only took it in stride but loved it, immediately embracing Bridget, who shares John's very dry sense of humor, so she and her dad do pretty well together. If I hadn't taken Bridget out, I think she would have become a prostitute or, worse, died." (The other infant was adopted by another family.) Her children, meanwhile, completely accepted their new sister. "They saw no difference; they never did. Nor did she, until recently."
(the last sentence has to do with the fact that in 2000 during his presidential run there were rumors that McCain fathered Bridget in an affair - there is more about that in the article)
McCain also talks about it in the DadMag article that was already posted.
Kim