When you type a profile, your browser asks this server, and this server asks Steam. The Steam Web API key is mine and stays here; your browser never sees it, and Steam never learns who was asking.
What comes back is what the profile already publishes to anyone: the display name, the avatar, the games, the hours, the achievements, and - where the profile publishes it - the trading cards sitting in its inventory, which is how a page can say what finishing a badge would cost instead of what a whole set costs. If a profile hides its game details, Steam returns nothing and the page says so. An inventory set to private is not read. Nothing private is unlocked here.
The answer is held in memory for fifteen minutes, so that a reload does not spend the key twice, and then it is gone. It is never written to disk. The one exception is a game's key art, cached as a plain JPEG at /art/<appid>.jpg - a picture of a game, with nobody's name on it.