So the idea is to approximate certain elements by other certain elements. Separable means that any element of the set can be approximated by a limited number of elements.
For example, with ##\mathbb{R}##, we can approximate an arbitrary element (for example ##e##) by their decimal representation. So we can approximate ##e## better and better by
2. ~2.7, ~ 2.71, ~ 2.718, ...
How do you use that analogy in non-metrizable separable spaces?
We can do exactly the same with any real number. We can do this in two dimensions too. For example, the couple ##(e,\pi)## can be approximated by
(2,3),~(2.7,3.1),~(2.71, 314),...
So separability actually is a very far-reaching generalization of the decimal representation. So basically, we have a countable number of "basis" elements, and then we succeed in approximating any element by the basis elements.
In the theory of Hilbert spaces, the analogy becomes even better with the existence of countably orthonormal bases. But I can only tell this if you know Hilbert spaces.
In a very general sense, you should see separable spaces and spaces which are "not too large" and where countable many terms suffice in many cases. In the same way that rational numbers can be used to describe real numbers.
It turns out that a very large portion of all spaces encountered "in nature" are separable. This is especially true in physics. This is good because it allows us to describe a physical system by countably many terms (like position, momentum, etc.) instead of dealing with uncountabilities.