I think "the" conceptual reason why an uncountable compact subset of $K\subset \mathbb{R}$ has the cardinality of the continuum is that it has a "canonical" map onto a space homeomorphic to $[0,1]$.

The complement of $K$ is a union of open intervals: a left ray, a right ray and at most countably many bounded ones.
The canonical map is the quotient map to the space obtained by gluing the end points of these bounded intervals in the complement
(it is easy to see that the equivalence classes of the equivalence relation thus obtained are at most countable).

This answers the question in the sense that this is a "simple reason why...". I am not saying that this is the simplest proof, as one still has to argue that the quotient space is homeomorphic to $[0,1]$ (a choice of such a homeo is certainly not canonical). There are various ways to do this. One option (again, not entirely elementary, but easy to explain) is to choose an appropriate probability measure $\mu$ on $K$ and define the map
$$ K \ni t \mapsto \mu(-\infty,t] \in [0,1].$$

The generic measure will do it for you, but if you actually want to construct one you can use the Lemma appearing in the body of the question (and take the non-atomic part of what you get).