real analysis - Understanding expansion on base $k$

There are some times when one might need to use expansion of real numbers on some base $k$. One example is when dealing with Cantor's set, one uses expansion of the numbers inside $[0,1]$ on base $3$. The point is that I simply can't understand these expansions. I'm asking here on the more general context of expansion on base $k$ in order to try to make the question more useful.

As I understood, expansion of a number $a\in \mathbb{R}$ in base 3 means to write

$$a = \sum_{n=1}^\infty \dfrac{a_n}{3^n} \quad a_n \in \{0,1,2\}.$$

In that setting, I imagine expansion of $a$ in base $k$ would mean to write

$$a = \sum_{n=1}^\infty \dfrac{a_n}{k^n} \quad a_n \in \{0,1,\dots, k-1\}.$$

Now what those expansions really mean? I simply can't understand, we are decomposing numbers as certain series. But what those series really mean? Why would anyone consider doing these expansions? What the coefficients $a_n$?

I believe this is related to decimal expansions, that is when we write a number $a = a_0.a_1a_2a_3\dots$ but I'm unsure how to make this connection rigorous. Also, I believe this would be true just for $k=10$, so for the other cases it would still be something hard to grasp.

In truth, I've seem quite a few times this being used in some proofs, the Cantor set being the most well-known example. But up to now I never understood correctly what these expansions are and how to work with them.