I know there are many questions and answers about this lemma. But I can not grasp the idea.
Let my countable collection of measurable sets will look like (-1/n,1/n) where n goes from 1 to infinity.
Is it countable collection of measutable sets? Yes.
Then there are so many numbers which do not belong to any of our sets. But lemma says that almost all x belong to at most finitely many of our sets.
Where is my mistake?
Following your example:
$$ \{A_n\}_{n=1}^\infty = \{ ( -\frac{1}{n},\frac{1}{n} ) \}_{n=1}^\infty $$
Then a given point $a$ in $\mathbb{R}$ such that $|a|\geq \frac{1}{n}$, then $a$ lies in $0$ intervals. You are right here.
Now see the Lemma, 'at most finetely' means that for each $x\in \mathbb{R}$ one can find a finite number $n\in\mathbb{N}$ such that:
$$ number\,\, of\,\,intervals\,\,x\,\,belongs\,\,to \equiv m \leq n $$
So, given the same $a$, we may use 1 and
$$ number\,\, of\,\,intervals\,\,a\,\,belongs\,\,to \equiv 0 < 1 $$