How can I clean the dirty tile pieces?

There's a cleaning machine located on the Central Plains.

You also need to make your own cleaning solution.

Or distill it.

There's a still on the wheel site.

It needs two ingredients.

One is near the still.

The item will fit in the hole on the still.

Put one of the mechanical seeds (growing in a field with small metallic
mounds) in the hole.

One of the other astronauts found the other ingredient you need.

He/she even labeled it for you.

It's the powder in the refrigerator at the tent site.  It will appear later
in the game if it's not immediately available.