Rain usually doesn't have dirt in it per se, but mainly will have tiny dust or pollen -- or even salt particles in it.
It's the dust particles that the water droplets stick to up in the clouds that create a raindrop. As the raindrop picks up more and more coatings of water from the surrounding water vapor, it'll be come larger and larger, until it's heavy enough to fall to the ground.
It's a good thing that we do have dust and other tiny particles in the air. If the air was super clean and didn't have any of those particles, it wouldn't be able to rain.