Let $W(\omega_1) := \{\beta \text{ is ordinal } \mid \beta < \omega_1\}$. Then why does $$ |W(\omega_1)| = |\mathbb{R}|? $$
Somehow my intuition breaks on this result. If we list out the first ordinals till $\omega_1$ we get the following list
$$1, 2, \dots, \mathbb{N}=\omega, \omega + 1, \omega +2, \dots, 2\cdot \omega, 2\cdot\omega + 1, \dots, k\cdot \omega, \dots, \omega\cdot\omega = \omega_1.$$
The result is now telling me that this list is uncoutable but why?
The set you call $W(\omega_1)$ is just $\omega_1$. For any ordinal $\alpha$ it is true that $\alpha = \{x\text{ ordinal}\mid x < \alpha\}$.
The cardinality $|\omega_1|$ is usually written $\aleph_1$, although in fact they are the same set. The cardinality $|\mathbb R|$ is usually written $\mathfrak c$. Whether $\aleph_1 = \mathfrak c$ is known as the continuum hypothesis, and famously it is independent of ZFC: within ZFC we can neither prove it true nor false (unless ZFC is inconsistent, in which case we can prove both). The only thing we can prove in ZFC is that $\aleph_1 \leq \mathfrak c$.
The list of ordinals you write down ends up at $\omega \cdot \omega$, usually written $\omega^2$. That ordinal is countable, as you correctly intuit, and in particular it is much, much smaller than $\omega_1$. In fact $\omega^2$ has the order type of the set of pairs of natural numbers, ordered lexicographically; this set is countable by much the same argument usually used to prove that $\mathbb Q$ is countable.