Breaking happens on the outer surface, because of tension building up: a crack initiates, causes further stress concentration, and failure. Now when you have two boards, once they are pressed together the friction between them makes the two boards behave a little bit like a single thicker board.
And in the case of a single thick board, the center line (which is actually the point where the top and bottom boards are touching) is the line of zero strain: above it the board is compressed and below it the board is stretched.
If there is no friction, or the boards are very far apart, then this analysis doesn't hold - but with sufficiently small spacing and rough surfaces it does.
This assumes identical boards; of course it is easy to choose the properties of the two boards so the top one is more elastic (or thinner, or has higher fracture strength)
Perhaps this diagram helps:
Notice how the top board is bent through a bigger curvature - that should mean more bending stress, and therefore more tension on the lower surface of the upper board. HOWEVER: if the top board is touching the bottom board (as in this diagram) then NECESSARILY the outer curvature of the top board must equal the inner curvature of the bottom board - and without friction, that means the upper board experiences more stress and should break first. But if there is friction with the surface of the bottom board, that makes the two boards "move as one", with the tension in the top board canceling the compression in the bottom board, and the tensile stress being transferred to the new "outer" surface (that is, the bottom of the lower board).