Your point about successive extensions of a basic number system is very well taken. We use the successive extensions
$$
\mathbb{N}\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\mathbb{R}
$$
to enable easier solution of problems in algebra and geometry. The Greeks had to do everything in terms of proportions referring to natural numbers alone, and this made things cumbersome. Sticking to ordered number systems, the next extension similarly is performed not for the sake of generality but rather to facilitate applications in infinitesimal calculus. Thus, once we complete the chain of extensions to
$$
\mathbb{N}\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\mathbb{R}\hookrightarrow{}^\ast\mathbb{R},
$$
we get greater facility at many levels, from elementary to research.
To give three simple examples, consider the following.
(1) The definition of the derivative becomes a finite procedure rather than an infinite limiting process. Thus the derivative of $y=f(x)$ is defined by $f'(x)=\text{st}\big(\frac{\Delta y}{\Delta x}\big)$ where $\Delta x$ is an infinitesimal $x$-increment, and "st" is the standard part function "rounding off" each finite hyperreal to the nearest real number.
(2) The definition of continuity of a function which in the A-track involves multiple alternations of quantifiers and are generally thought to be confusing to students, can be replaced by what was essentially Cauchy's definition: a function $f(x)$ is continuous if an infinitesimal $x$-increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.
(3) Terry Tao has spoken and written extensively about the expressive power of Robinson's framework for analysis with infinitesimals and its utility in research; e.g, in this 2017 publication in Discrete Analysis.
The claim in the video you link to is incorrect; to be honest, no video making such a claim should be used as a reference for technical topics. There is no substantive sense in which $\mathbb{R}(x)$ is related to hyperreals.
The most that can be said is that $\mathbb{R}(x)$ is an ordered (well, fine, orderable) field with infinitesimal elements - that is, a non-Archimedean field. Hyperreal fields, however, are much more than merely non-Archimedean fields. A hyperreal field must have a "high degree of similarity" to $\mathbb{R}$ itself, in a particular technical sense. $\mathbb{R}(x)$ demonstrably lacks this, most obviously since it does not satisfy the property "$\forall u\ge 0\exists v(v^2=u)$." So we don't even have $\mathbb{R}(x)\equiv\mathbb{R}$, let alone that $\mathbb{R}(x)$ is a hyperreal field (see below)!
To someone new to the subject of nonstandard analysis, this may feel like uncharitable hair-splitting. I want to emphasize as strongly as possible that the difference between mere non-Archimedeanness and hyperrealness is huge. Remember that the whole point of hyperrealness is that a hyperreal field should let you "transfer" results to the standard real numbers $\mathbb{R}$, so that infinitesimal methods can still provide real results. The fact that this is possible at all is really neat and awesome, and ignoring precisely what makes hyperrealness special in the interest of a superficial gain in accessibility is a pedagogical disservice.
It may help at this point to see the (most common in my experience) precise definition of hyperreal-ness, in order to contrast it with $\mathbb{R}(x)$. Let $\Sigma$ be the expansion of the language of fields by a symbol for every finite-arity relation on the reals, and let $\mathcal{R}$ be the reals construed as a $\Sigma$-structure in the obvious way. A hyperreal field is then just a proper elementary extension $\mathcal{F}$ of $\mathcal{R}$.
Note that strictly speaking a "hyperreal field" isn't a field but rather an expansion of a field (compare $\mathcal{R}$ with the much-less-complicated $(\mathbb{R};+,\times,0,1)$. This is a bit messy, so we describe $\mathcal{F}$ as a field equipped with a third-order operation ${}^*$ which assigns to each finite-arity relation $A$ on $\mathbb{R}$ an extension $^*A$ on the underlying set of $F$ which satisfies some simple properties. This amounts to the same thing as the above, but is arguably more convenient in two significant ways: pdagogically this approach is more accessible to non-logicians, and technically it lets us pay more careful attention to exactly how much transfer is needed for a given application.
Best Answer
There are a couple of Monthly articles that give reasonably accessible introductions. See
Wm. Hatcher. Calculus is Algebra, AMM 1982.
D.H. Van Osdol. Truth with Respect to an Ultrafilter or How to make Intuition Rigorous. AMM, 1972..
For a much more comprehensive introduction to ultraproducts see
Paul Eklof. Ultraproducts for Algebraists, 1977.