Someone once told me that numbers such as $\sqrt{2}$ have a unique mathematical meaning:
$$\sqrt{2}=\lim\{1,1.4,1.41, ... \}$$
While I understand that this might be sufficient for a formal mathematical definition, it still fails to provide me with intuition. The numbers within the curly braces seem to be approaching a number that has a fixed, definite value, but the string of digits still goes on forever. Additionally, because there is no way of knowing all of the digits of $\sqrt{2}$, it seems strange that we can reason with this number so easily. Is there a more intuitive interpretation of what irrational numbers mean?
The statement you've written down is a theorem, which states that given any $x \in \mathbb R \setminus \mathbb Q$, there exists a sequence of rational numbers $(a_n)_{n=1}^\infty$ such that: $\lim_{n\to \infty}=x$.
If one wants to talk about the intuition behind that statement, there's a (beautiful) observation to be made first: if $x \in \mathbb Q$, then the decimal expansion of $x$ will eventually become periodic, or terminate - if there is a "point after which" the expansion is composed entirely of zeros. This is a consequence of Dirichlet's pigeonhole principe.
Moreover, if $x \in \mathbb R \setminus \mathbb Q$, then the decimal expansion of $x$ will not terminate, and will never become periodic.
Now,an intuition for irrational numbers, based on this theorem, could go on as follows:
If $x$ is an irrational number, its decimal expansion does not terminate and never becomes periodic. However, there exists a sequence of rational numbers $(a_n)_{n=1}^\infty$ that converges to $x$ - and the decimal expansion of every individual term in that sequence is an approximation to the decimal expansion of $x$, which gets better and better the "further" you go along this sequence.
This is a soft way of saying that, as $n$ gets larger and larger, the approximation gets closer and closer to the actual decimal expansion of $x$, and the precision with which this happens can be as good as you want it to be - exact to within $1,000$ digits after the dot, or $1,000,000$, or $100,000,000$ and so on. Hope this helps.