I know what it means, but can't really see the reasoning of this phrase. Anyone with an insight?
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According to wikipedia, quoting OED, it was first used in baseball around 1880.
This is just speculation, but another meaning of check is "to stop":
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.