Why do they say "Cheque Please"?

Because years ago, the little pad that the waitress would write the order on, was called a 'book of Checks.' I do not know if they are still called that, but when I waited tables, that is what they were called.

In earlier American history, many small paper notes involving financial matters were referred to as "cheques," which later evolved into "checks" spelling-wise. "Travelers checks," a "checking account/checkbook," etc. are good examples, and the bill one received for dining out has retained this nomenclature into the present day. So, in the U.S. at least, they're actually saying, "check, please," and although the difference in spelling is obviously not apparent in the pronunciation, if you're writing about this subject to Americans, they may not immediately understand your meaning.

Restaurant or "guest check" is the term used in the U.S.

It is called a "check" because it is an itemized list for the patron to "check" (verify) that the items he/she ordered are listed with prices, and then the patron can agree on what he/she is being charged.

They say "Cheque" because that's what they call a "bill" in America.

Because, in the Statrs the term for "the bill" is "the check". We get that concept from German, "die Rechnung, bitte" which literally means 'the reckoning, please'. In order to arrive at the correct bill, we need to check it first.

It's 'check' and is just the name given to a restaurant bill.

Because to say "Bill please!" is too cliche.And saying "What`s the damage?" might not be understood by Chinese, Persian or Ethiopian waiters (depending in what restaurant you eating out.)

I think it means check please, as in bill