Not an answer, but too long for a comment: here is how I would explain (not prove) that $L\models CH$ to someone who has not seen Lowenheim-Skolem (actually, I would really prefer to just show them Lowenheim-Skolem - it's easy and beautiful! - but suppose for some reason I can't).
There's a proof of a related claim which does not use Lowenheim-Skolem (well, it uses the key idea). We can define a "baby $L$" for second-order arithmetic: for $\eta\in ON$, we define $M_\eta$ recursively as
$M_0=(\omega, \{\})$,
$M_\lambda=(\omega, \bigcup_{\alpha<\lambda}M_\alpha$),
$M_{\alpha+1}=\Pi^1_2-Def(M_\alpha)$,
where $\Pi^1_2-Def(N)$ is the set of subsets of the first-order part of $N$ which are $\Pi^1_2$-definable over $N$.
Now the "$L\models CH$"-like claim is that this construction terminates at level $\omega_1$ and results in at most $\omega_1$-many reals; that is, $$M_{\omega_1}=M_{\omega_1+1}=\bigcup_{\eta\in ON} M_\eta \quad\mbox{and}\quad \vert \mathbb{R}^{M_{\omega_1}}\vert\le\omega_1.$$
Note that the second claim follows immediately from the first, since at each countable stage we only add countably many reals; so suppose $r$ is a real definable over $M_{\omega_1}$ via some formula $\varphi$ (WLOG, using no parameters) - so $n\in r$ iff $M_{\omega_1}\models\varphi(n)$. For each $n\in\omega$, let $\mathcal{F}_n: \mathbb{R}^{M_{\omega_1}}\rightarrow\mathbb{R}^{M_{\omega_1}}$ be the function defined as:
if $M_{\omega_1}\models\varphi(n)$, then $\mathcal{F}_n$ is a Skolem function for $\varphi(n)$; and
if $M_{\omega_1}\models\neg\varphi(n)$, then $\mathcal{F}_n$ is the constant function always spitting out a counterexample real.
Of course, we don't want to use the phrase "Skolem function" here, but the beauty of a $\Pi_2$-sentence is that it's easy to define what a Skolem function is - we don't have to talk about Skolemization.
Now we can easily show that there is some $\alpha<\omega_1$ which is closed under every $\mathcal{F}_n$. Thus $r$ is defined by $\varphi$ over $M_\alpha$, and so is already in $M_{\alpha+1}$.
At this point there are two gaps: how do we replace "$\Pi^1_2$-definable" with "definable;" and how do we turn this into an "internal" (to $L$, that is) argument? (Note that it makes no sense to talk about this construction within second-order arithmetic, so the argument above is decidedly external.)
At this point, I'd introduce Skolemization to handle the first point (which is easy in light of the proof above), and say that what's needed for the second point is the ability to "make local truth-definitions" inside $L$ (note that these truth-definitions are the crucial step in the condensation lemma itself). This still leaves the serious work of building those truth-definitions, and checking that everything works, undone; but at this point they should have a sense of why the claim is true at least.
Again, this isn't anything close to a proof that $L\models CH$; but hopefully it would give some of the intuition with less of the technical difficulty.
