The boys have gashes because they are competitors in the kite contest. What do gashes on the hands have to do with flying kites? Well, the string on the kites is prepared in a special way, soaked in a mixture that includes ground glass. The glass is meant to stay on the string so the string of one kite might cut another kite out of the sky. The person who has the last kite flying is the winner, which means that each participant wants to cut all the other kites out of the sky. Imagine what it feels like to have glass-coated string run through your fingers when you fly a kite! It is bound to cause cuts in the palm. Even when I was a child and flew kites, the wind would pick up the kite and make it rise quickly. If the string was wrapped in my hands, I remember that it hurt. And there was no glass coating on my string. If you have ever flown a kite, you might recall this pain!
The boys all have these gashes on their fingers from fighting kites. The strings of the kites are covered with ground up glass and glue so that they will be able to cut the strings of the other kites. The sharp strings end up cutting their fingers because the boys' fingers come in contact with the strings when they are doing their kite fighting.
This kite fighting is where the title of the book comes from. When the older people would fight kites in a more serious way, they would pay boys to chase down the kites whose strings got cut. Those boys were the kite runners.
In the book "The Kite Runner" the first part and most of the story takes place in Afghanistan. The winter in Kabul is a time of school shutting down for several months due to the icy season. The winter months also brings a time for kite flying. Every year there would be a kite flying tournament.
To prepare for the tournament meant that the children would buy supplies and build their own kite. They would work on the kites and pull the strings across their fingers as they set up the line. The result was scars across their fingers caused by the line.
The gashes were horizontal lies cut across their fingers from a whole winter of practice flying the kites. The children would gather together and inspect each others gashes and scars when they returned to school.
Page 50 has this part of the story.