late Old English scoru "twenty," from Old Norse skor "mark, notch, incision; a rift in rock," also, in Icelandic, "twenty," from Proto-Germanic *skur-, from PIE root *sker- (1) "to cut."
The connecting notion probably is counting large numbers (of sheep, etc.) with a notch in a stick for each 20. That way of counting, called vigesimalism, also exists in French: In Old French, "twenty" (vint) or a multiple of it could be used as a base, as in vint et doze ("32"), dous vinz et diz ("50"). Vigesimalism was or is a feature of Welsh, Irish, Gaelic and Breton (as well as non-IE Basque), and it is speculated that the English and the French picked it up from the Celts. Compare tally (n.).
The prehistoric sense of the Germanic word, then, likely was "straight mark like a scratch, line drawn by a sharp instrument," but in English this is attested only from c. 1400, along with the sense "mark made (on a chalkboard, etc.) to keep count of a customer's drinks in a tavern." This sense was extended by 1670s to "mark made for purpose of recording a point in a game or match," and thus "aggregate of points made by contestants in certain games and matches" (1742, originally in whist).
From the tavern-keeping sense comes the meaning "amount on an innkeeper's bill" (c. 1600) and thus the figurative verbal expression settle scores (1775). Meaning "printed piece of music" first recorded 1701, said to be from the practice of connecting related staves by scores of lines. Especially "music composed for a film" (1927). Meaning "act of obtaining narcotic drugs" is by 1951.
Scoreboard is from 1826; score-keeping- from 1905; newspaper sports section score line is from 1965; baseball score-card is from 1877.