elementary set theory - Understanding the proof for $2^{aleph_0} > aleph_0$

I'm trying to understand how $2^{\aleph_0} > \aleph_0$. I was reading through this sketch of the proof, but don't quite understand how they show that $\mathrm{card}((0,1)) = \mathrm{card}(\mathcal{P}(\mathbb{N}))$. Is there a different way of explaining this? Or maybe a different way of explaining the whole proof? I'm just trying to wrap my head around this, so any help is appreciated!