In A Beginner's Guide to Modern Set Theory [page 48], the author says:
[Cantor] did prove that every closed uncountable subset of $\mathbb R$ has cardinality $2^{\aleph_0}$...
... but I cannot find the proof anywhere, and it doesn't seem trivial. Can someone help me, either by pointing me to a proof, or proving it directly?
A perfect set is a set which is equal to the set of its limit points. (In particular, it's closed.)
Any nonempty perfect subset of $\mathbb{R}$ has cardinality $2^{\aleph_0}$.
The Cantor Bendixson Theorem states that any closed subset of $\mathbb{R}$ can be written (uniquely) as the disjoint union of a countable set and a perfect set. Hence an uncountable closed set contains a nonempty perfect set, and hence has cardinality $2^{\aleph_0}$.
(The above two statements about $\mathbb{R}$ work for Polish spaces more generally.)