Old English bisig "careful, anxious," later "continually employed or occupied," cognate with Old Dutch bezich, Low German besig; no known connection with any other Germanic or Indo-European language. Still pronounced as in Middle English, but for some unclear reason the spelling shifted to -u- in 15c.
The notion of "anxiousness" has drained from the word since Middle English. Often in a bad sense in early Modern English, "prying, meddlesome" (preserved in busybody). The word was a euphemism for "sexually active" in 17c. Of telephone lines, 1893. Of display work, "excessively detailed, visually cluttered," 1903.