Lemma 10.2: There exists a well-ordered set $A$ having a largest element $\Omega$, such that the section $S_\Omega$ of $A$ by $\Omega$ is uncountable but every other section of $A$ is countable.
(1) What is being claimed in the lemma here? That we can divide/partition (English words equivalent to break) a well-ordered set into parts/sections - an uncountable and a countable part? I am not able to understand the punchline here.
(2) Is there an example for a set $B$ which I can use to understand the proof? Is $X^\omega$ where $X=\{0,1\}$, an uncountable set, well-ordered in the dictionary order (doesn't seem so though...)?
Re: your first question, I think the stumbling point is terminological: "section" is a technical term in Munkres, referring to sets of the form $\{x: x<a\}$ for some element of the ordering $a$. The introduction of this term is easy to overlook at first, which makes statements about sections seem either mysterious or trivial.
I think it may help to rephrase the lemma as follows:
As to your second question, unfortunately this is a rather complicated object. For example, the usual axioms of set theory cannot decide whether it has the same cardinality as $\mathbb{R}$. (For example, it's a good exercise to show that $\{0,1\}^\omega$ with the lexicographic order is not well-ordered and has lots of uncountable sections.)
It can be "explicitly constructed," but that explicit construction is rather abstract; the discussion here is relevant (although it refers to the version of $A$ with the "top element" removed). Understanding this object is an important and meaningfully difficult step in the process of learning "intermediate" set theory; for a first course in point-set topology, it's really only touched on and is used more than it is understood (FWIW I think there's a vague analogy with the role of Zorn's lemma in abstract algebra courses).