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The following sentence appears in this video at around 7 min:
教科書の中でキツツキは木に穴をあけるとありましたが、一秒間のうちに何回どんどんどんどんってできるでしょうか？
The follwing sentence appears in this answer:
"メニューには売り切れとある" (menu says [it is] sold out.)
I want to know what do 「と」 and 「ある」 mean in these sentences?
In this Japanese dictionary, 「とある」 got a meaning:
according to; reads; reading; is written; says; states. Abbreviation, See also と書いてある
but, how does it abbreviate from と書いてある?
～とある can be reworded using ～と書いてある in most of the times, but it doesn't mean that it is the abbreviation of the latter. "The sign reads 'STOP'." is a kind of peculiar grammar too, so to speak, but not an abbreviation of "The sign can be read 'STOP'."
This と is quotative. The so-called "quotative" と isn't just for closing quotations; its real function is to convert an arbitrary language snippet into an adverbial clause in a sentence. Thus the phrase ～と modifies the verb ある.
教科書の中でキツツキは木に穴をあけるとありました
"in the textbook, there was literally 'woodpeckers make holes in trees'"
= "the textbook had (a content) literally 'woodpeckers make holes in trees'"
= "the textbook (we saw) says 'woodpeckers make holes in trees'"
Incidentally, the past tense ありました does not mean that it no longer says in this context, but they no longer see the textbook, as you see in the translation above.
