The gizzard has two halves split by a hard cartilage-like bone. It's easily recognizable by a diagonal line running through the gizzard. Cut along this bone like material but not all the way through.
Split the two halves. Inside will be the grit that the bird uses to help digestion and grind seeds.
Take the spilt open gizzard and clean it in a bowl of water. Make sure to get all the grit out. DO NOT do this under the sink. It will clog your pipes very quickly. Also when you are done do not throw this water down the sink, just dump it in your backyard.
Split the gizzard in two now. Start trimming all the edges and silverskin material and anything that doesn't look like meat.
Now remove the grinder plate, the hard cartilage-like material. Slice along side it and try not remove too much meat with it.
Repeat on the other half. Remove the silverskin. You can start by taking the tip of the knife cut some of the silverskin off and then peel it off. The gizzard should look like a roundish pink meat.
There are gizzard recipes throughout the web so find one that looks appealing. If you're worried about eating innards try a fried recipe to make it easier to start. I use the same marinade I use for ducks and cook it with the bird. If you have concerns remember a gizzard is just meat. It is harder but it does not have any gaminess or ironiness that other innards have, so it is an easy entry meat for the nasty bits. Texturally though it is harder than flesh but its quite edible. Give it a shot.