What is the meaning of delta squared?
I hate to break it you, but they're hiding the math from you that is really going on behind the scenes.

It is very common in situations where the students are not expected to know calculus, that wherever a derivative would be needed, to replace, for example, the $dx$ with a $\Delta x$. This is because for many simple physical systems, taking the derivative results in simply measuring the average change in a property with respect to some other property.

For instance, consider a system where you know the position of some particle and how this position changes with time. If you want to know the velocity of that particle at any given time, you would take the derivative with respect to time, $$v=\frac{dx}{dt}$$but, in many cases you might be only interested in the average velocity and could thus skip the derivative part and write that $$v=\frac{\Delta x}{\Delta t}=\frac{x_2-x_1}{t_2-t_1}$$

But, and now we're getting to your question, let's take it one step further and say we wanted to know the acceleration of that particle. This would be the derivative of velocity which is also written as the second derivative of position with respect to time. Thus, we would write $$a=\frac{d^2x}{dt^2}$$Notice how similar this looks to your question about $\Delta^2$ if you simply replace the $d$ with $\Delta$.

Again, if we decided we were only interested in the average acceleration, then we would write $$a=\frac{\Delta^2 x}{\Delta t}$$
The reason why they say that $$\frac{\Delta(\frac{\Delta a}{\Delta b})}{\Delta b}=\frac{\Delta^2 a}{\Delta b^2}$$is again because that is true of derivatives. This would be written as $$\frac{d}{dt}(\frac{dx}{dt})=\frac{d^2x}{dt^2}=a$$

So, to summarize, the symbol $\Delta$ is often used in situations where you care about a change from one state to another (as with a change in energy) or when you only care about the average of some quantity.

As to why they are able to write$$\frac{\Delta^2 a}{\Delta b^2}=\frac{a_3-a_2}{\Delta b}$$ I'm not certain about. I would have to know what this dealt with specifically. My guess is that the quantity $a_3$ is somehow related to the second derivative of $a$ with respect to $b$ and they did some algebra behind the scenes.

Hope that helps.