Openly; without any trickery.
It is sometimes suggested that the board in question is the deck of a ship and that this comes from the seafaring practice of concealing pirates below decks (below board) in order to lull victim ships into a false sense of security. The opposite, 'above board' was considered to symbolize openness and fairness.
There's no evidence to support that derivation and it seems clear that this term originated in the gaming community. If card players keep their hands above the table (board) they can be seen to be playing fairly.
Beaumont & Fletcher's The Custom of the Country, 1616, includes this line:
"Yet if you play not fair play and above board too, I have a foolish gin here, I say no more." [Laying his hand upon his sword]
Given the phrase 'above board', we might expect there to be a 'below board' or 'beneath the board'. Neither of these is recorded, although 'under board' was used briefly. Sir Christopher Heydon used it in In Defence of Judiciall Astrology, 1603:
"After the fashion of iugglers, to occupie the minde of the spectatour, while in the meane time he plaies vnder board."
The phrase that won out as the converse of 'above board' was clearly 'under hand', now usually written as a single word. This dates back to the 17th century too, for example, in this line from Cyril Tourneur's The Atheist's Tragedie, 1611:
"He does it under hand."