Generalization Error Definition
There exists somewhere in the world a distribution $D$ from which you can draw some samples $x$. The notation $x \sim D$ simply states that the sample $x$ came from the specific distribution that was noted as $D$ (e.g. Normal or Poisson distributions, but also the possible pixel values of images of beaches). 

Say you have some ground truth function, mark it as $c$, that given a sample $x$ gives you its true label (say the value 1). Furthermore, you have some function of your own, $h$ that given some input, it outputs some label.

Now given that, the risk definition is quite intuitive: it simply "counts" the number of times that $c$ and $h$ didn't agree on the label. In order to do that, you (ideally) will 


go over every sample $x$ in your distribution (i.e. $x \sim D$).
run it through $c$ (i.e. $c(x)$) and obtain some label $y$.
run it through $h$ (i.e. $h(x)$) and obtain some label $\hat{y}$.
check if $y \neq \hat{y}$. If so, you add 1 to your count (i.e. $1_{h(x) \neq c(x)}$ - that notes the indicator function)


Now last thing to note is that I wrote above "count", but we don't really care if the number is 500 or 100, we care for the relative number of mistakes (like 40% or 5% of the samples that were checked were classified differently). That is why it is noted as expectancy ($\mathbb{E}$).

Let me know if that was clear enough :-)