\begin{equation} \beta(x_0) = \int_{-\infty}^{\infty}\alpha(x_0-x)\phi(x)dx\end{equation}
This is a convolution.
Convolutions are quite useful things to work with and show up all over the place.
If you know signal processing, the pointwise multiplication of a fourier transform of two functions is equal to the fourier transform of the convolution of two functions. And the fourier transform is its own inverse.
In signal processing, you use the fourier transform to extract the frequency components of an input function. Pointwise multiplication in the "frequency domain" corresponds to convolution in the time domain; so you can use convolution to do a "band pass" filter on a function (say, drop all components above a certain frequency).
This is all relatively academic, but the point is that convolutions are common and quite useful in a wide variety of places.
Now, if we imagine the Dirac Delta as a function we can convolve with another function such that when we do it, we extract the original function out.
\begin{equation} \int_{-\infty}^{\infty}\delta(x-x_0)\phi(x)dx = \phi(x_0)\end{equation}
You'll notice how this looks much like a convolution. Such forms are well studied and you can use other functions in place of the Dirac delta there and get reasonably interesting results.
For this to be "true", we just need delta to be a function that is zero everywhere except at 0, and its integral from negative epsilon to positive epsilon be 1. Then the math of convolution gives us that the result of the convolution is \phi(x_0)
.
\begin{equation} \int_{-\infty}^{\infty}\delta(x_0)\phi(x)dx = \beta(x_0)\end{equation}
Here, if we use the same imaginary Dirac delta, we end up with $\beta(x_0)$ equal to infinity at $x_0=0$, and equal to zero everywhere else.
This isn't $\phi(x_0)$ as we want.
The $x-x_0$ term is non-zero unless $x=x_0$, and zero if $x=x_0$. In a sense, this forces the entire "sampling" of $\phi$ to occur at $x=x_0$ and zero contribution to occur elsewhere.
To make this formal without using extended definitions of "function", we state that instead of the Dirac delta being one function, it is instead a limit of functions, and the convolution is the limit of the convolutions.
We take any ordered set of functions whose limit off 0 is 0, and whose integral from negative epsilon to positive epsilon limits to 1 from below for any epsilon.
For example, $\delta_i(x) = i/2$ if $|x|<1/i$ and 0 otherwise. Or a myriad of other options.
Now we take this:
\begin{equation} \int_{-\infty}^{\infty}\delta_i(x_0-x)\phi(x)dx = \beta_i(x_0)\end{equation}
This gives us a series of functions $\beta_i$, generated by a traditional convolution. The limit of the $\beta_i$ is $\phi$ regardless of what form $\delta_i$ take.1
Now, the equation you showed was $x-x_0$ instead of my convolution $x_0-x$. Well, the $\delta_i$ I used are symmetric around 0 -- so $\delta_i(x-x_0)=\delta_i(x_0-x)$.
In short, what is going on is that convolution is useful, the "pretend" Dirac delta treated as a convolution equation is often useful, and what is "really" going on can be interpreted as a limit of convolution of functions for which the equation stands in for.
Now this being mathematics, when we have a useful concept like Dirac delta, we go off and redefine what the symbols mean to make the Dirac delta a "generalized function" and make the notation we want to use "just work" without it "being shorthand for what is really going on", because all practical math is just shorthand anyhow.
Finally, $\delta(x-x_0)(\phi)$ just looks like sloppy notation. They are using $\delta$ as a kind of macro to me. I don't see the use of it, really.
I'd want $\delta_\phi(x_0)$ or $\delta(\phi)(x_0)$ or even $\delta_{x_0}(\phi)$ or $\delta(x_0)(\phi)$ or or $\delta * \phi$ equal to $\int_{-\infty}^{\infty}\delta_i(x_0-x)\phi(x)dx$ (where $*$ is convolution)
1 My fourier analysis is a bit rusty, I'd believe there are pathological functions that don't behave well when convoluted yet satisfy the requirements I placed on the $\delta_i$. So, I'll hedge my bets and say "sufficiently nice" functions instead.