We know we can over board by the induction hypothesis.
To cover a board we build one as follows. (See below).
4 smaller squares are . When we bring them together we get a big square. Now we ask, "is it possible oto tile this big square"? And the answer is, yes! But how do we show it? The small red square represents a missing peice in this big square. WLOG we will assume it is in the upper left square. Now by the induction hypothesis we can tile the lower left, upper right, lower right squares with one tile removed. But not just any tile, we pick a tile that meets in the center, shown by blue squares. Now, we can tile the LL,UR,LR square because one tile is removed. We can tile the UL square because only one tile is removed by induction hypothesis. And we can tile the middle blue square because they are arranged in a shape of a tromino. Thus, we can tile the entire thing!