How do proponents of the perpetual virginity of Mary view Matthew 13:53–57? Matthew 13:53–57 reads:


  53 When Jesus had finished these parables, he moved on from there. 54 Coming to his hometown, he began teaching the people in their synagogue, and they were amazed. “Where did this man get this wisdom and these miraculous powers?” they asked. 55 “Isn’t this the carpenter’s son? Isn’t his mother’s name Mary, and aren’t his brothers James, Joseph, Simon and Judas? 56 Aren’t all his sisters with us? Where then did this man get all these things?” 57 And they took offense at him. (NIV)


I have read that proponents of the doctrine claim that the word translated as "brothers" in Matthew 13:55 does not necessarily imply biological relationship (http://www.catholic.com/blog/matt-fradd/jesus-had-brothers). Looking at the verses though, I find it difficult to imagine why the word translated as "brothers" would be anything but biological. Why would people who are in Jesus' hometown (verse 53) and who also used the word "sisters" (verse 56) use "brothers" in a way that is not biological?
The link you quote (which I will also link to here) explains the view of Catholics on this passage in some detail. The article explains how they interpret the passage, and gives examples of other places where 'brother' does not imply coming from the same womb. If it was a normal usage of the time, then there is no reason why people of Jesus' hometown would not have used it of course, and 'sisters' presumably has similar usage.

Not all Christians agree with the interpretation, and if you are not a Catholic you are free to hold a differing view.