What is $F(0)$ is "dc" component in the context of image processing? It has always been said that $F(0)$ is the "DC component" in fourier transform. However, I don't get what it means to say that $F(0)$ is "DC" in the context of image processing. 

The zero in this case just meant zero frequency, and hence no changes isn't it? Then in the context of image processing, how should I imagine the picture to be like at $F(0)$? What exactly is that "DC" component?
The zero frequency of an image DCT is the mean gray value of the pixels of the input image (for graylevel images).
$F(0)$ is not an image: it is a single coefficient.