(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain there.
You can find the slides here, under "Recent results about the Continuum Hypothesis, after Woodin". (In true Bourbaki fashion, I heard that the talk was not well received.)
Roughly, Woodin's approach shows that in a sense, the theory of $H(\omega_2)$ decided by the usual set of axioms, ZFC and large cardinals, can be "finitely completed" in a way that would make it reasonable to expect to settle all its properties. However, any such completion implies the negation of CH.
It is a conditional result, depending on a highly non-trivial problem, the $\Omega$-conjecture. If true, this conjecture gives us that Cohen's technique of forcing is in a sense the only method (in the presence of large cardinals) required to establish consistency. (The precise statement is more technical.)
$H(\omega_2)$, that Dehornoy calls $H_2$, is the structure obtained by considering only those sets $X$ such that $X\cup\bigcup X\cup\bigcup\bigcup X\cup\dots$ has size strictly less than $\aleph_2$, the second uncountable cardinal.
Replacing $\aleph_2$ with $\aleph_1$, we have $H(\omega_1)$, whose theory is completely settled in a sense, in the presence of large cardinals. If nothing else, one can think of Woodin's approach as trying to build an analogy with this situation, but "one level up."
Whether or not one considers that settling the $\Omega$-conjecture in a positive fashion actually refutes CH in some sense, is a delicate matter. In any case (and I was happy to see that Dehornoy emphasizes this), Woodin's approach gives strength to the position that the question of CH is meaningful (as opposed to simply saying that, since it is independent, there is nothing to decide).
(2) There is another approach to the problem, also pioneered by Hugh Woodin. It is the matter of "conditional absoluteness." CH is a $\Sigma^2_1$ statement. Roughly, this means that it has the form: "There is a set of reals such that $\phi$", where $\phi$ can be described by quantifying only over reals and natural numbers. In the presence of large cardinals, Woodin proved the following remarkable property: If $A$ is a $\Sigma^2_1$ statement, and we can force $A$, then $A$ holds in any model of CH obtained by forcing.
Recall that forcing is essentially the only tool we have to establish consistency of statements. Also, there is a "trivial" forcing that does not do anything, so the result is essentially saying that any statement of the same complexity as CH, if it is consistent (with large cardinals), then it is actually a consequence of CH.
This would seem a highly desirable ''maximality'' property that would make CH a good candidate to be adopted.
However, recent results (by Aspero, Larson, and Moore) suggest that $\Sigma^2_1$ is close to being the highest complexity for which a result of this kind holds, which perhaps weakens the argument for CH that one could do based on Hugh's result.
A good presentation of this theorem is available in Larson's book "The stationary tower. Notes on a Course by W. Hugh Woodin." Unfortunately, the book is technical.
(3) Foreman's approach is perhaps the strongest opponent to the approach suggested by Woodin in (1). Again, it is based in the technique of forcing, now looking at small cardinal analogues of large cardinal properties.
Many large cardinal properties are expressed in terms of the existence of elementary embeddings of the universe of sets. These embeddings tend to be "based" at cardinals much much larger than the size of the reals. With forcing, one can produce such embeddings "based" at the size of the reals, or nearby. Analyzing a large class of such forcing notions, Foreman shows that they must imply CH. If one were to adopt the consequences of performing these forcing constructions as additional axioms one would then be required to also adopt CH.
I had to cut my answer short last time. I would like now to say a few details about a particular approach.
(4) Forcing axioms imply that $2^{\aleph_0}=\aleph_2$, and (it may be argued) strongly suggest that this should be the right answer.
Now, before I add anything, note that Woodin's approach (1) uses forcing axioms to prove that there are "finite completions" of the theory of $H(\omega_2)$ (and the reals have $\aleph_2$). However, this does not mean that all such completions would be compatible in any particular sense, or that all would decide the size of the reals. What Woodin proves is that all completions negate CH, and forcing axioms show that there is at least one such completion.
I believe there has been some explanation of forcing axioms in the answer to the related question on GCH. Briefly, the intuition is this: ZFC seems to capture the basic properties of the universe of sets, but fails to account for its width and its height. (What one means by this is: how big should power sets be, and how many ordinals there are.)
Our current understanding suggests that the universe should indeed be very tall, meaning there should be many many large cardinals. As Joel indicated, there was originally some hope that large cardinals would determine the size of the reals, but just about immediately after forcing was introduced, it was proved that this was not the case. (Technically, small forcing preserves large cardinals.)
However, large cardinals settle many questions about the structure of the reals (all first order, or projective statements, in fact). CH, however, is "just" beyond what large cardinals can settle. One could say that, as far as large cardinals are concerned, CH is true. What I mean is that, in the presence of large cardinals, any definable set of reals (for any reasonable notion of definability) is either countable or contains a perfect subset. However, this may simply mean that there is certain intrinsic non-canonicity in the sets of reals that would disprove CH, if this is the case.
(A word of caution is in order here, and there are candidates for large cardinal axioms [presented by Hugh Woodin in his work on suitable extender sequences] for which preservation under small forcing is not clear. Perhaps the solution to CH will actually come, unexpectedly, from studying these cardinals. But this is too speculative at the moment.)
I have avoided above saying much about forcing. It is a massive machinery, and any short description is bound to be very inaccurate, so I'll be more than brief.
An ordinary algebraic structure (a group, he universe of sets) can be seen as a bi-valued model. Just as well, one can define, for any complete Boolean algebra ${\mathbb B}$, the notion of a structure being ${\mathbb B}$-vaued. If you wish, "fuzzy set theory" is an approximation to this, as are many of the ways we model the world by using a probability density to decide the likelihood of events. For any complete Boolean algebra ${\mathbb B}$, we can define a ${\mathbb B}$-valued model $V^{\mathbb B}$ of set theory. In it, rather than having for sets $x$ and $y$ that either $x\in y$ or it doesn't, we assign to the statement $x\in y$ a value $[x\in y]\in{\mathbb B}$. The way the construction is performed, $[\phi]=1$ for each axiom $\phi$ of ZFC. Also, for each element $x$ of the actual universe of sets, there is a copy $\check x$ in the ${\mathbb B}$-valued model, so that the universe $V$ is a submodel of $V^{\mathbb B}$. If it happens that for some statement $\psi$ we have $[\psi]>0$, we have established that $\psi$ is consistent with ZFC. By carefully choosing ${\mathbb B}$, we can do this for many $\psi$. This is the technique of forcing, and one can add many wrinkles to the approach just outlined. One refers to ${\mathbb B}$ as a forcing notion.
Now, the intuition that the universe should be very fat is harder to capture than the idea of largeness of the ordinals. One way of expressing it is that the universe is somehow "saturated": If the existence of some object is consistent in some sense, then in fact such object should exist. Formalizing this, one is led to forcing axioms. A typical forcing axiom says that relatively simple (under some measure of complexity) statements that can be shown consistent using the technique of forcing via a Boolean algebra ${\mathbb B}$ that is not too pathological, should in fact hold.
The seminal Martin's Maximum paper of Foreman-Magidor-Shelah identified the most generous notion of "not too pathological", it corresponds to the class of "stationary set preserving" forcing notions. The corresponding forcing axiom is Martin's Maximum, MM. In that paper, it was shown that MM implies that the size of the reals is $\aleph_2$.
The hypothesis of MM has been significantly weakened, through a series of results by different people, culminating in the Mapping Reflection Principle paper of Justin Moore. Besides this line of work, many natural consequences of forcing axioms (commonly termed reflection principles) have been identified, and shown to be independent of one another. Remarkably, just about all these principles either imply that the size of the reals is $\aleph_2$, or give $\aleph_2$ as an upper bound.
Even if one finds that forcing axioms are too blunt a way of capturing the intuition of "the universe is wide", many of its consequences are considered very natural. (For example, the singular cardinal hypothesis, but this is another story.) Just as most of the set theoretic community now understands that large cardinals are part of what we accept about the universe of sets (and therefore, so is determinacy of reasonably definable sets of reals, and its consequences such us the perfect set property), it is perhaps not completely off the mark to expect that as our understanding of reflection principles grow, we will adopt them (or a reasonable variant) as the right way of formulating "wideness". Once/if that happens, the size of the reals will be taken as $\aleph_2$ and therefore CH will be settled as false.
The point here is that this would be a solution to the problem of CH that does not attack CH directly. Rather, it turns out that the negation of CH is a common consequence of many principles that it may be reasonable to adapt in light of the naturalness of some of their best known consequences, and of their intrinsic motivation coming from the "wide universe" picture.
(Apologies for the long post.)
Edit, Nov. 22/10: I have recently learned about Woodin's "Ultimate L" which, essentially, advances a view that theories are "equally good" if they are mutually interpretable, and identifies a theory ("ultimate L") that, modulo large cardinals, would work as a universal theory from which to interpret all extensions. This theory postulates an $L$-like structure for the universe and in particular implies CH, see this answer. But, again, the theory is not advocated on grounds that it ought to be true, whatever this means, but rather, that it is "richest" possible in that it allows us to interpret all possible "natural" extensions of ZFC. In particular, under this approach, only large cardinals are relevant if we want to strengthen the theory, while "width" considerations, such as those supporting forcing axioms, are no longer relevant.
Since the approach I numbered (1) above implies the negation of CH, I feel I should add that one of the main reasons for it being advanced originally depended on the fact that the set of $\Omega$-validities can be defined "locally", at the level of $H({\mathfrak c}^+)$, at least if the $\Omega$-conjecture holds.
However, recent results of Sargsyan uncovered a mistake in the argument giving this local definability. From what I understand, Woodin feels that this weakens the case he was making for not-CH significantly.
Added link: slides of a 2016 lecture by Woodin on Ultimate L.
