what does "on the landing"mean?
“Your tea,” she said, indicating a steaming mug.
“Great, thanks. Just give me a moment,” he said, and he left to pee in the bathroom on the landing. As he zipped up his fly, he caught sight of himself in the mirror, crumpled-looking and unshaven. Not for the first time, he consoled himself that his hair looked the same whether brushed or unbrushed.

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Answer

"The bathroom on the landing" refers to the bathroom which has it's doorway opening off the landing or flat area at the top or bottom of a flight of stairs.

Definition of landing - a noun - 'a level area at the top of a staircase or between one flight of stairs and another, as in "he ran across the landing to his bedroom"'