I like to explain that Chess is a game of war. The king is the commander. If he is captured, the war is over.
"Checkmate" literally means "the king is dead." (Expand on sheiks and elephants and rocs if they like that kind of thing...)
Chess is also a polite game. When you attack the king, you are required to warn your opponent to protect his king. You say "check", which means "king."
It's so polite that we don't allow a player to put himself into check, and if a player in check makes a move that doesn't get him out of check, we'll let him try again until he finds a move that gets him out of check.
Actually, checkmate means "the king is surprised" or "the king is left helpless." The following is from etymology.com:
Checkmate (14th c.) comes via Old French eschec mat from Persian shāh māt ‘the king is left helpless’ ... From the very specific chess sense there developed more general applications such as ‘attack’, ‘arrest’, ‘stop’, ‘restrict’, and ‘verify’. Among these in the 18th century was ‘token used as a counterfoil for verifying something, such as an amount’. As check this survives mainly in American English (as in ‘hat-check’)
Жэня