Why is a restaurant bill called a "check" (as in "Check, please!")?
This is a good question, and the general references (Online Etymology Dictionary, etc.) are not helpful in getting a definite answer. That is, they give the general etymology of check (it arose from chess, and "All the other senses seem to have developed from this one"), but not of this particular "restaurant bill" sense. Even the Oxford English Dictionary does not explain it; however its classification is suggestive. It says:
14. A means to ensure accuracy, correctness, security from fraud, etc.: as
†a. The counterfoil of a bank bill, draft, etc.
b. A token, usually a memorandum of receipt, a ticket, or piece of metal […] used for the purpose of identification, or as evidence of ownership or title: given, e.g. to the owner of luggage on a railway (as in U.S.), or to one who temporarily leaves luggage, cloaks, […] etc., etc.
c. A restaurant bill. Chiefly U.S.
1869 A. D. Whitney … I let her settle for the dinner checks.
1910 ‘O. Henry’ … Through an arched opening‥you thrust your waiter's check and the money.
1916 Variety … Inspectors‥ordered drinks and paid their check just before one.
1937 R. Stout … I‥found the waitress and got my check from her.
(Looking under cheque, it seems to have arisen as "a differentiated spelling" of check in sense 14a, so the modern meaning of cheque (or American [bank] check) is different from 14c.)
So to answer your question, we don't know exactly, but the heading under which the OED has put the word suggests that they think it came from "a means to ensure (check) correctness". It's not entirely clear what the restaurant bill is meant to check (perhaps the diner checks that he/she is being asked for the correct amount). This listing is still not as definite as an explicit pronouncement on the origin of this sense, so something concrete from a real expert would be useful.
In the etymonline entry for check you can find:
c.1700 "a token used to check against loss or theft" (surviving in hat check)
Maybe it was first used to call the waiter to bring you your hat (coat, cane, etc.) and the bill; in the "olden" days, when it was customary to wear hats, it was not unusual for restaurants to have hat racks at which people would leave their hats.
Another possibility is that the guest wanted to check the bill, before paying for it.
No references to support either theory.
According to The Online Etymology Dictionary, the usage of the word "check" comes from chess, as in "check mate", when the game is over:
early 14c., "A call in chess noting one's move has placed his opponent's king in immediate peril," from O.Fr. eschequier "a check at chess" (also "chess board, chess set"), from eschec, from V.L. *scaccus, from Arabic shah, from Pers. shah "king," the principal piece in a chess game (see shah; also cf. checkmate). When the king is in check a player's choices are limited. Meaning widened from chess to general sense of "adverse event" (c.1300), "sudden stoppage" (early 14c.), and by c.1700 to "a token used to check against 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 exchequeur. Meaning "pattern of squares" (c.1400) is short for checker. Checking account is attested from 1923, Amer.Eng.
I assume that either as Unreason said, this could be asking for a "hat check", or more simply, just asking for the end of the meal, since check is the end of a chess game.
It's also extremely possible that it's related to the usage of the word "check" in American English to refer to a form of payment ("cheque" in British English)