Why do they say planets and their moons are parts of the solar system?

Answer 1

Because the sun (Sol) has them all trapped in its gravitational field. All the planets revolve around the sun and so all the planet's satellites revolve around the sun, too.

There are quite a few more objects that are part of Sol's gravitational field- comets, asteroids etc.

Answer 2

Planets and their moons are considered part of the solar system because they orbit around the Sun, which is the central star of the solar system. They interact with each other through gravitational forces and together form a cohesive system within our solar system.