Being in shape and having a background in some kind of sport can be a benefit when starting to climb, but, it is not a requirement. Almost anyone, regardless of how fit there are and their weight, can climb.
While you climb, you will probably start to feel muscles that most refer to as "muscles they have never felt before and didn't know existed". This is because climbing utilizes muscles that most people usually do not use that often, at least not in the way climbing uses them. This is especially true for people who do not come from a fitness background of some kind.
Climbing in itself can help you get in shape, but getting in shape also depends on what you do outside the climbing gym or away from the rocks. It has to do with how you treat your body, how often you climb or exercise, if you stretch and work on preventing/treating injuries, and of course on your diet.
