I'm self-learning some measure theory. During the following lecture ( I had included the appropriate starting time inside the link, but timespan is 7:50-9:00):
https://youtu.be/llnNaRzuvd4?t=470
the Professor Claudio Landim states that if we take $\Lambda = \mathbb{R}|_{\sim}$ where $\sim (x, y) : y - x \in \mathbb{Q}$ then $\Lambda$ is uncountable. The lecturer explains that if indeed $\Lambda$ was countable, then every element in $\mathbb{R}$ could be represented by element of $\Lambda$ (some equivalence class) and by element of this equivalence class. Couldn't it? Why not? I cannot understand it.
Your notations are not so clear. $$\mathbb R/_\sim=\{[x]\mid x\in \mathbb R\},$$ and $$[x]=\{y\in \mathbb R\mid y-x\in \mathbb Q\}.$$
So, what you want to prove is that $\Lambda :=\mathbb R/_\sim$ is uncountable. Suppose $\Lambda $ is countable, i.e. $\mathbb R/_\sim=\{[x_n]\}_{n\in\mathbb N}$ for some $x_n$. In one hand, if $x\in \mathbb R$, then $[x]=x+\mathbb Q$ is countable. In the other hand, if $\Lambda $ is countable, then $\mathbb R$ is a countable union of countable set, and thus $\mathbb R$ would be countable, which is a contradiction.