transformation for non-constant variance? This is from my textbook

I don't understand what does the content in red mean, for example, what does $y^2_i \infty  \sigma_i$ mean? How can we tell the relationship between $y^2_i$ and $\sigma_i$ by looking at the residual plot?(I don't actually know the symbol between $y^2_i$ and $\sigma_i$, so I use $\infty$ instead). 
Another question is, for the first one, why is not $E[y^2_i]=a+bx$ rather than $E[1/y]=a+bx$?
 A: 
for example, what does $y^2_i\propto σ_i$ mean? 

The part in the red box looks looks backward to me. I think what they're actually trying to say there is "If $\sigma_i\propto y_i^2$" (the proportionality was fine but they make it sound like it's being driven by the wrong thing is all - we're basically estimating how the spread changes as the mean changes (note that the plot is vs $x$ rather than $y$, so arguably it should be "If $\sigma_i\propto E(y_i)^2$" but I won't harp on that issue for now). Note that the display to the right is $r_i$ vs $x_i$ and see that $\sigma_i$ is the population standard deviation of the $i$-th error term (which terms we estimate by residuals), and since the spread is increasing quadratically in $x_i$ it is at least potentially possible for it to be proportional to $y_i^2$ (it requires that point of zero variance to be at a particular place on the x-axis).
Note that you need the $y$'s to all be positive for this to make sense.
If that proportionality ($\sigma_i=k y_i^2$) were the case, then via a Taylor expansion, $Var(1/y_i)$ will be approximately $1/E(y_i)^4 \sigma_i^2 = k^2$, a constant; this choice of a transformation to make the variance approximately constant is typically referred to as a variance-stabilizing transformation.
There are a number of posts on site relating to Taylor expansions and variance stabilizing transformations.
If spread was proportional to $y_i$ rather than its square, you might consider taking logs and if it was proportional to $\sqrt{y_i}$ you might consider taking square roots.
(There are alternatives to transformation, however, that may well be more suitable in many cases)
Similarly for the second red box; taking $y^2$ would again (to first order) stabilize the variance when the relationship between variance and mean is reciprocal for the same reason.

Beware, however -- there's no clear reason to expect the relationship on the transformed scale to be linear. Indeed if it were linear before transformation (and it must have been the case - at least to a good approximation - for the residual plot to be useful in the way we're being asked to use it) then it won't generally be linear after transformation.

How can we tell the relationship between $y^2_i$ and $σ_i$ by looking at the residual plot?

Look at how the spread changes on average with the increase in $x$ (and hence $y$).
