Stomping ground (chess)

Or from elsewhere

soheil1"Notice how he[=Black] can move only on the 7th and 8th ranks, whereas White has the first six ranks as his stomping ground."

I have no idea what you are trying to say.

A "stomping ground" is (eg.) a place where you used to live and know the area very, very well. It is familiar territory to you.

I can suggest several definitions.See what applies please:

White is going to stay a long time there.
White possesses there
White has the sole right to that area
White is resident there
...

White well understands the first six ranks, and deftly assembles his pieces in strategic positions there.

What about 'he completely dominate the first 6 rank'?

The sentence does not say that directly or indirectly, but it might be a conclusion one would reach.

Sorry, the whole sentence means
White has made the first 6 ranks his stomping ground
or just that
the first 6 ranks are White's stomping ground?
or something else?

If it is talking about one specific game then: White controls the first 6 ranks. He can use those positions to attack Black.

If it is talking about White's skills as a player, then White specializes in controlling the first 6 ranks and using these in planning his strategy.