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.

Put that in the other slot on the still.

Finally, turn the wheel to get the still running.

The cleaning fluid you made will be on the right side of the still.

Press the button to release some of the fluid from the still.

You can put the dirty tile pieces on the shelf just below the spout.

But the shelf itself is dirty, and you need to clean that first.

Press the button once before you put any tiles on the shelf.  That will clean
the shelf for you.

Now, you can put the tiles on the shelf and clean them by pressing the button
again.