The "full answer" version of this question has been asked elsewhere on MSE, e.g. here. However, as far as I can tell the question "What is a good hint for showing that there is an uncountable chain of sets of naturals" has not been asked here, and I think that is a worthwhile separate question since it gets at techniques for developing intuition about "large" families of sets.
It's not hard to show that no well-ordered chain of subsets of $\mathbb{N}$ can be uncountable. For example, if $\alpha$ is an ordinal and $(A_\eta)_{\eta<\alpha}$ is a well-ordered increasing chain of sets of naturals, let $$x_\eta=\min(A_{\eta+1}\setminus A_\eta).$$ Then the map $\eta\mapsto x_\eta$ is an injection from $\alpha$ into $\mathbb{N}$; since the latter is countable, $\alpha$ (and hence our chain itself) must be countable as well. (The argument for well-ordered decreasing chains is basically identical.)
But chains don't have to be well-ordered, and this argument breaks down when we drop that hypothesis. After playing around with the problem, one eventually comes to the conclusion that an uncountable chain of sets of naturals may in fact exist, despite how weird that seems.
So how do we find such a chain?
The key idea is to look for a similar-flavored situation which we already understand. In this problem, we have a countable "starting set" $\mathbb{N}$ and we're trying to use it to "generate" somehow an uncountable family of related objects (the sets in our chain). So let's begin with a modest goal:
Can you think of an example of building an uncountable set from a countable "starting set"?
- Hint: don't look too far - you've worked with this one a lot already!
Here's the answer I have in mind:
Going from $\mathbb{Q}$ to $\mathbb{R}$.
Now we want to somehow view this example as building a family - hopefully a chain! - of sets. The version you're familiar with likely doesn't do this immediately:
It's probably via equivalence classes of Cauchy sequences - and these are much more than just sets of rationals!
However, there is another way to approach the construction which does:
Dedekind cuts: a real $r$ can be thought of as the corresponding set $\{q\in\mathbb{Q}: q<r\}$.
Now do you see how to "lift" this idea over to the problem you care about?
There's another example to consider, incidentally, which is less familiar but extremely important:
The infinite binary tree $2^{<\omega}$ - that is, the set of all finite binary strings.
The uncountable set "built from" this countable set is
the set of paths through the tree - that is, the set of all infinite binary sequences.
Turning this into a chain of subsets of the countable starting set is a bit harder; can you see a way to associate
a set of nodes on the tree - or finite binary strings
to each
infinite binary sequence?
- HINT: as in the previous example, it may help to think of an ordering - and then we can talk about "the set of things to the left" ...
Namely, to each infinite binary sequence $f$ we associate the set of finite binary strings which are to the left of $f$: that is, the set $$\{\sigma\in 2^{<\omega}: \sigma\not\prec f\mbox{ and }\sigma(k)<f(k)\mbox{ for $k$ the first point of difference}\}.$$
This example is worth understanding since it's related to a lot of other important examples. For instance, once understood it leads to a quick-and-easy construction of an uncountable antichain of sets of naturals - indeed, an uncountable almost disjoint family (= all intersections are finite) set of naturals:
Associate to an infinite binary sequence $f$ the set of initial segments of $f$.
