Why do only certain orbitals exist in an atom?

Because the electrons in an atom move about as waves, they can follow only certain allowed orbits that we call orbitals. This limitation is equivalent to the case of a violin string—it can only vibrate at certain frequencies. If you try to make a violin string vibrate at the wrong frequency, it won't do it. That's because the string vibrates in a wave-like manner and only certain waves fit properly along the strong. Similarly, the electron in an atom "vibrates" in a wave-like manner and only certain waves fit properly around the nucleus.