I know what it means, but can't really see the reasoning of this phrase. Anyone with an insight?
A ticket given to a spectator at an outdoor event providing for a refund of his or her entrance money or admission at a later date, should the event be interrupted by rain.
According to wikipedia, quoting OED, it was first used in baseball around 1880.
[In] baseball ... when a game was rained out, those who had tickets for that game were given a rain check which they could redeem at another game. ... The use of the term as early as 1884 gives some indication as to the popularity of baseball in the U.S. even at that time: 'The heavy rain yesterday threw a damper over local operations. At each of the parks the audience had to be content with three innings and rain checks.' (St. Louis Missouri Post-Dispatch 26 May [1884?])
This is just speculation, but another meaning of check is "to stop":
1. An action or influence that stops motion or expression; a restraint: Heavy rains were a check on the army's advance.
2. The condition of being stopped or held back; restraint: kept my temper in check; holding agricultural pests in check with sprays.
3. An abrupt stop in forward movement or progress; a halt.
"In order to understand how the falling rain checks the wave motion,"
"abundance of rain checks the propagation of cholera"
So, it could also be said that rain checks play. Perhaps the issuing of the paper checks (cheques, tickets) was also a pun on this meaning.
Edit: here it is in the Baltimore American of Dec 12, 1884, in a short column headed BASEBALL REGULATIONS about "Changes made in the Rules of the American Association". If the phrase has been used in official regulations, it's likely to have been used verbally for some time beforehand.
When clubs issue rain checks, they shall not be required to pay the $65 guarantee to visiting clubs.