1530s, "supplying of meals, food and lodging," from board (n.1) in its extended sense of "food" (via notion of "table"). Boarding-school is from 1670s; boarding-house attested from 1728.
Entries linking to boarding
"piece of timber sawn flat and thin, longer than it is wide, wider than it is thick, narrower than a plank;" Old English bord "a plank, flat surface," from Proto-Germanic *burdam (source also of Old Norse borð "plank," Dutch bord "board," Gothic fotu-baurd "foot-stool," German Brett "plank"), perhaps from a PIE verb meaning "to cut." See also board (n.2), with which this is so confused as practically to form one word (if indeed they were not the same word all along).
In late Old English or early Middle English the sense was extended to include "table;" hence the transferred meaning "food" (early 14c.), as "that which is served upon a table," especially "daily meals provided at a place of lodging" (late 14c.). Compare boarder, boarding, and Old Norse borð, which also had a secondary sense of "table" and an extended sense "maintenance at table." Hence also above board "honest, open" (1610s; compare modern under the table "dishonest"). A further extension is to "table where council is held" (1570s), then transferred to "leadership council, persons having the management of some public or private concern" (1610s), as in board of directors (1712).
"Bow to the board," said Bumble. Oliver brushed away two or three tears that were lingering in his eyes; and seeing no board but the table, fortunately bowed to that.
Meaning "table upon which public notices are written" is from mid-14c. Meaning "table upon which a game is played" is from late 14c. Meaning "thick, stiff paper" is from 1530s. Boards "stage of a theater" is from 1768.