What you are seeing on your griddle is called "heat tint." When stainless steel is made, a small amount of chromium is added to protect the iron in the steel from rusting. When the steel is heated, as in the case of your griddle, the chromium is forced to the surface and forms a layer of protective oxide. That's the discoloration we see in this picture. This discoloration is non toxic and does not leach into your food.
Is there any way to get rid of this? Yes and by "yes" I mean maybe. You can try using something like Bar Tender's Friend to remove the discoloration. But, I would strongly caution you from introducing chemicals to your cooking surface for obvious reasons, especially since this discoloration is going to happen every time the griddle is used.
Like I said, the discoloration is perfectly harmless and over time and with use, your entire griddle will eventually take on this discoloration, including the sides and corners.