> Real numbers are those which can be represented on a number line. As per definition

That isn't really the definition of real numbers - it's just a way of thinking about them. Real numbers are usually defined in terms of sets or sequences of rational numbers (fractions) - you can google "Dedekind cut" or "Cauchy sequence" if you want the full details, but they're maybe a bit beyond ELI5.

> but if we don't know their exact precise value then how can we plot it?

On a real-life number line, we don't really know the exact position of any number, since we don't have any perfect measuring devices. We can use real numbers to model reality in approximate terms, but they don't correspond exactly to reality. For example, we know that there are infinitely many real numbers between 0 and 1, but are there are infinitely many points between two distinct positions in space? Nobody really knows.