busy (adj.) Look up busy at Dictionary.com
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.
busy (v.) Look up busy at Dictionary.com
late Old English bisgian, from busy (adj.). Related: Busied; busying.