There are three ways to approach this. 1) Attach your L Brackets to the wall, then balance your game boards on top. NOT very professional but it does work and doesn't damage potentially valuable old boards.
2) Attach your L brackets to the wall (opposite way that they're attached in the photos, "upside down") and glue your boards to the brackets! Choose a strong adhesive like E600 or hot glue. You can also use strong double-sided tape (3M makes double-sided tape that won't hurt the walls, I have it pictured in the last photo, the green/white roll) to attach them.
3) Instead of just placing your boards atop the L brackets you can nail the top of the boards to the wall, attach them to the wall with double sided tape, or use a picture hanging kit on the backs of the boards. You can attach the base to the L brackets with more glue or even attach a piece of wood for more stability.
Whatever method you choose, I'd make sure to stage these with lighter-items only! No breakables. This is not the spot to perch your bowling ball collection, clearly. I made my shelves from G.I. Joe and Mad Magazine game boards I found at the flea market last weekend. The project cost $6.00 total! (The L brackets were .25 cents each on clearance at IKEA.) It really balances out the other corner of my "office" space for the moment (until I install a wall-o-shelves, can't wait) and goes with my playful home aesthetic. Fun times, now to make a million more!
Dig through your closets to find game boards with missing pieces or keep an eye out for them, they're always popping up at flea markets and thrift stores!