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"Check out this article"
Where did this term come from and why those choice of words? I understand it perfectly semantically, but when you think about it it doesn't make sense, check out seems to mean to exit, e.g. “I checked out of the hospital, my toe was fine”.
We "check" with someone before coming and going in various circumstances. You mention a hospital; other examples include a hotel, a grocery store, a secure apartment complex. The expression comes from your identity, credentials, or authorization being examined ("checked") on your way in or out.
The expression has been variously extended to encompass other meanings nowadays: "I tried to get Joe to help, but he seems checked out these days" meaning that Joe is disengaged.
Your original question, before and after editing, exhibits yet another example of how subtle English can be; these two sentences mean very different things:
I checked out that hotel; it was awful!
I checked out of that hotel; it was awful!
The first implies you merely examined the hotel; the second implies you were staying there and decided to leave.
Your first use case ("Check out this article") appears to be a variation on the theme of "check against forgery" mentioned below:
check (n.1) Look up check at Dictionary.com c. 1300, "a call in chess noting one's move has placed his opponent's king (or another major piece) in immediate peril," from Old French eschequier "a check at chess" (also "chess board, chess set"), from eschec "the game of chess; chessboard; check; checkmate," from Vulgar Latin *scaccus, from Arabic shah, from Persian shah "king," the principal piece in a chess game (see shah; also compare checkmate (n.)). Also c. 1300 in a generalized sense, "harmful incident or event."
When the king is in check that player's choices are severely limited. Hence, "sudden stoppage" (early 14c.), and by c. 1700 to "a token of ownership used to check against, and prevent, loss or theft" (surviving in hat check) and "a check against forgery or alteration," which gave the modern financial use of "bank check, money draft" (first recorded 1798 and often spelled cheque), probably influenced by exchequer. Checking account is attested from 1897, American English. Blank check in the figurative sense attested by 1849. Checks and balances is from 1782, perhaps originally suggesting machinery.
checkout has a different origin for hotels, hospitals, etc:
checkout (n.) Look up checkout at Dictionary.com 1944, from check (v.1) + out (adv.). Originally "training given to a pilot for using a specific aircraft;" hotel sense is from 1958.)
