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According to the Oxford English Dictionary it came from the Old Norse skor, which mean notch, tally or twenty. It grew out of this definition of score which began in Middle English:

A drawn line, orig. one constituting a boundary (chiefly fig. in out of score, beyond the limit, excessively, unreasonably); a stroke, a mark. ME.

By the early 18th century it came to mean:

A written or printed piece of a composition showing all the vocal and / or instrumental parts arranged one below the other.

According to Grove's Dictionary of Music, scoring refers to drawing measure lines across several staffs, connecting them. This makes it obvious all the staffs are parts, to be read by different musicians simultaneously, and are not one continuous line to be read sequentially. This then became a verb implying arranging: to score a piece meant writing all the individual parts, as opposed to simply composing a single tune or melody line.