In simple words: What does Woodin's $(*)$ axiom and Martin's maximum state? I have just read an article in "Spektrum der Wissenschaft (2021)", a german journal that tries to communicate scientific topics to the non-expert in a comprehensible way, named "Unendlichkeit - Wie viele reelle Zahlen gibt es?" (which should translate to: Infinity - How many real numbers are there?).
The article explains a method called "forcing", which was introduced by Cohen in 1963, that shows that the continuum hypothesis is independent from ZFC. According to the article, the method of forcing introduces problems. As Hugh Woodin puts it:

This new technique creates an ambiguity in our universe of sets. You create all these virtual possibilities. This raises the question of how to judge which mathematical reality we are in.

The article does not explain this ambiguity and I could not grasp it. I accepted the need for "closing" the gap as the article calls it.
However, while the forcing method is explained in detail, the article continues with two very confusing statements. The first one tries to explain Martins Maximum, a forcing axiom (an axiom that tries to close the introduced ambiguity) named after Donald A. Martin, in a single sentence:

Thus, any object that can be generated by forcing becomes a real part of mathematics if the procedure satisfies a certain condition.

The article does not explain what "a real part of mathematics" means or what this magical condition is. Woodin introduced a different axiom, called $(*)$ that sets the cardinality of the real numbers to $\aleph_2$ without explaining the actual axiom. A few moments later it is stated

The biggest difference between $(*)$ and Martin's maximum is in the foundations on which they are built. While the latter refers to ZFC, Woodin's axiom is based on a model universe of sets that satisfies the nine ZF axioms and the axiom of determinacy AD instead of the axiom of choice C.

I have tried to research the above, but at first glance it seems incomprehensible to the non-expert and all in all and the article has raised more questions than it has answered.
Question:
In relatively simple words, what is Martin's maximum and what is the $(*)$ axiom?
 A: At an ultra-high level, what's going on is this. Forcing provides a class of methods for modifying models of $\mathsf{ZFC}$ in controlled ways. We can use these methods to prove independence results: by showing how from an arbitrary "starting model" of $\mathsf{ZFC}$ we can produce a model of $\mathsf{ZFC+\neg\varphi}$, we establish that $\mathsf{ZFC}$ can't prove $\varphi$ as long as $\mathsf{ZFC}$ is consistent in the first place (implicitly invoking the soundness and completeness theorems).
Now there's a sense in which forcing amounts to "filling in a gap" in the universe. Consequently we might look for "completeness" principles, saying that no additional forcing is possible. If formulated naively these are outright inconsistent, but it turns out that more careful ways of posing them are quite interesting and amount to strengthenings of the Baire category theorem. For example, $\mathsf{MA_{\aleph_1}}$ is equivalent to the statement that whenever $\mathcal{X}$ is a topological space satisfying a couple conditions, the intersection of $\aleph_1$-many dense open subsets of $\mathcal{X}$ is nonempty. Martin's Maximum, $\mathsf{MM}$, is a much more technical variation on this same theme, and $\mathsf{MM^{++}}$ is a further elaboration on that. Ultimately the slogan is something like the following:

In a precise sense, $\mathsf{MM^{++}}$ is the strongest possible extension of the Baire category theorem, and can be interpreted as saying that the universe is complete with respect to certain types of forcing constructions.

Now what about Woodin's axiom $(*)$? This is a highly technical elaboration on the same idea as the above. We lose, in my opinion, some obvious appeal; I find $\mathsf{MM^{++}}$ much more compelling than $(*)$ at first glance. However, what we gain (in the presence of large cardinals anyways) is an incredible canonicity phenomenon: modulo $(*)$, any statement about subsets of $\omega_1$ cannot be altered by forcing. This is ultra-desirable, at least some of the time, and so it's natural to be excited that it follows from $\mathsf{MM^{++}}$:

Very broadly speaking, it turns out that - in the presence of large cardinals, anyways - an "ultra-Baire-category-theorem" hypothesis more-or-less settles the theory of $\mathcal{P}(\omega_1)$.

This should be considered in analogy with the fact that large cardinal hypotheses on their own more-or-less settle the theory of $\mathcal{P}(\omega)$ (= projective absoluteness).
