The Hebrew word for the corner of their garment is kanaf; a word, as with most Hebrew words, has double meaning. kanaf can mean corner or fringes and on one occasion it is translated wings:
Malachi 4:2 - ...The sun of righteousness will come will rise with healing in its wings.
One reason for the double meaning is because when the man would lift the corners of his outer garment (which would happen when a blessing occurred) the image you would see would look like wings of a bird. So over time the two ideas sort of went together. When you think of verses about being in the shadow of God's wings, you're literally in the protecting embrace of your Father.
The verse from Malachi above was seen in Jesus day as an image used to describe what Messiah will be like when He comes, they took the word used for "wings" and gave it it's original meaning - corner. So they took it literally that when Messiah comes He will have healing on the corner of his garment. Does this remind you of a story from the gospels? Do you remember the story of the woman suffering with an issue of blood. With what we just spoke of, let's reread this passage.
You can find the passage here Matt 9, Mark 5, Luke 8
The story has three main characters in it, Jesus, the Sick Women and His Disciples. The location is on a road leading to the home of a prominent leader of a synagogue, Jesus leading his disciples and a crowd surrounding Him as He went. As He was on His way a woman who had been suffering for twelve years, a sickness that made her unclean and seperated from society, learned that Jesus was passing by and she remembered from the scriptures that if this man was Messiah, than there would be healing waiting for her in His kanaf. She took her chances, straining through the crowd, reached for the corners of his garment and was healed.
The story tells us that Jesus realized what had happened, that He began to look around for who "touched" Him. The disciples were puzzled as there were many people around Jesus, it could have been anyone who nudged or bumped into Him. But Jesus added, "Someone touched me, I know that power has gone out from me."
When the woman heard Jesus asking who touched Him she knew she had to respond, so she did. Jesus had only kind words for her, "Daughter, your faith has healed you, Go in Peace."
One thing I learn from this story is that the woman had a thorough education, we don't know anything else about her but we can glean from this that her family studied the scriptures intently. In the first century, the boys of the family would memorize the first five books of Moses, if they did well they would continue studying and the best of the best would have most if not all of what we call the Old Testament memorized. Girls on the other hand, would most likely learn the Psalms and Proverbs, it was customary for the women to lead the singing during their worship service, primarily from the Psalms. The point being, that the idea of healing in the wings/corner of the Messiah's garment was from one of the minor prophets (Malachi 4:2), one of the books that not even some of the men would have studied but for this woman, because her family loved the book, she found healing in Messiah.
What aspects of Messiah Jesus do we fail to see because we don't read intently into His story? Could it be that our 21st century eyes cause us to miss the deep wells of our Gospels? I would suggest that the answer is yes, and that if we want to be like Jesus (which is what discipleship is all about) than we need to do our best to see the context behind these ancient words. It is a challenge but the reward could be life-changing.