A shadow is an area where direct light from a light source cannot reach due to obstruction by an object. It occupies all of the space behind an opaque object with light in front of it.
I have tried the same experiment keeping my two fingers as the rods you showed, I didn't get such bright thing at the intersection point. I tried the same thing taking a semi-transparent pen and a non-transparent pen. At the intersection point, I observed the brightness. If you are asking the explanation for this case, the non-transparent pen produces a shadow, the semi-transparent pen has some light rays passing through it. Thus, the region appears little bright at the intersection point. You can see in the above picture stolen from wikipedia, where the shadow from transparent region appears little bright, and no such brightness is seen at the opaque part.
I didn't believe initially with olin lathrop answer. But when I saw the below picture, I really started to believe.
Actually there are white spaces in between the dark squares, but we are seeing dark spots in between (in your case you can exchange colors). It is the ghost image, an optical illusion. It is caused when our cone receptors lose sensitivity due to overstimulation. When I tried in home, my cone receptors were not over stimulated, so I didn't see bright spots at the intersection points.