I don't understand the parts highlighted in green. I understand that the supports imply that X and Y are not independent but not how the graph shows this graph. I'm a bit confused by all aspects of the graph to be frank.
$Y=X^2 + Z$ to avoid any confusion.
Without getting too deep into the mathematical definitions, consider the following informal argument.
The notion of independence of two random random variables is intended to capture the intuitive meaning of independence: given the value of one of the random variables, you do not know anything more about the other than you did before, that is, whatever value $X$ might take on does not influence the value of $Y$. So, in your case of $Y = X^2+Z$, we know that $X^2$ has value in $[0,1)$ while $Z$ has value in $(0,0.1)$. Thus, the value of $Y$ can be anywhere in $[0, 1.1)$; indeed, if you had never heard of $X$ and $Z$ but were merely observing $Y$, that would your conclusion: $Y$ takes on values in $[0,1.1)$.
But now suppose that you know that $X$ had value $0.5$. Then, from the knowledge that $Y = X^2 + Z$, that $Y$ must have value in $(0.25,0.35)$. Thus knowing the value of $X$ tells you something that you did not know before about $Y$. No longer can $Y$ range from $0$ to $1.1$; it must necessarily be restricted to the picayune range $(0.25,0.35)$. Hence, $X$ and $Y$ are not independent.
Since informal arguments are often eschewed in favor of analytical verifications, here is a simple test for dependence that requires only some elementary notions from geometry. If the joint distribution of $(X,Y)$ does not have support that is a rectangle with sides parallel to the axes, then the random variables are dependent. (More knowledgeable folks: please hold your fire, I am trying to keep it as simple as possible to get the essential point across). In your example, the banana-shaped hand-drawn figure cannot be construed to be a rectangle, and so this eyeball test says that $X$ and $Y$ are dependent. In particular, laboriously calculation of $f_X(x)$ and $f_Y(y)$ and then checking whether it is true that $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ for all $x$ and $y$ is totally unnecessary. That $(X,Y)$ is uniformly distributed on this region is true in your example, but not relevant to the issue of independence which can be answered in the negative simply by looking at the banana.
Don't apply this argument in the opposite direction: the support being a rectangle does not prove independence; it merely suggests that it might be true that $X$ and $Y$ are independent random variables, but to prove that they are indeed independent requires more work.
The region has four boundaries: two parabolas (one is 0.1 higher than the other), and two vertical line segments on the left and right sides. Choosing a point uniformly at random in this region is equivalent to choosing $X$ uniformly at random in $[-1,1]$, squaring, then adding a random number chosen uniformly at random in $[0,0.1]$.
