Rock climbing is a very arm-heavy sport, as it mostly utilizes muscles responsible for pulling and holding. The muscles most used for these are your back muscles and your arms. Just like with lifting weights, these muscles can tear and get damaged, which causes the pain you may feel, but it's also what causes them to grow and get stronger.
Rock climbing requires you to pull much of your body weight, and to keep you on the wall and close to it. This means that your muscles, especially your arm muscles, with be very heavily used for this, and in ways that you may not be used to, especially if you are a beginner rock climber.
Your shoulders, all the way down to your forearms, are used to help you move from hold to hold, and to help you grip the holds and stabilize yourself on the wall. This means that you will be pulling much of your body weight on with your arms, and you will most likely be doing it in ways that will cause your muscles to go through the same process they would go through when lifting weights. They will get damaged, which is what will cause them to hurt, and it is also what will cause them to grow and get stronger. Eventually, as you progress with your climbing, they will hurt less.
