In the olden days, people didn't have box springs and the like; the mattress was held off the floor by rope that was threaded back and forth through holes in the bed frame. Over time these ropes would start to sag, and you'd have to tighten the ropes.
Enter the bed wrench. It is simply a small stick or narrow board with a couple of pegs sticking out to either side at the top end, and a deep notch cut into the bottom end. The notch would be placed over a loop of rope, twisted to pull it tight; while maintaining tension on the rope, you would then move the wrench down to the next loop. Once you've pulled as much slack as possible out of the system you re-tied the terminal knot, replaced the mattress, and were ready to sleep.
Needless to say, these are pretty much useless these days. Bed wrenches are still sold as collectors items, and an old one in good condition can sell for over a hundred dollars. I have not yet seen one that was particularly aesthetically pleasing, so the value presumably lies in being able to tell the story behind that weird thing on your mantelpiece.