How does a computer interpret real numbers?
They represent continuous quantities with discrete approximations.  Mostly, this is done with floating point, which is analogous to scientific notation.  Essentially, they work with something like $1.xyz\times 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).

It's also possible to work with some irrational numbers directly.  For example, you could create yourself an object called "$\sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(\sqrt{2})^2=2$.  So then you could compute $$(\sqrt{2}-1)(\sqrt{2}+1) = (\sqrt{2})^2 + \sqrt{2} - \sqrt{2} - 1 = 2 - 1 = 1$$
exactly, as an algebraic fact that's not susceptible to rounding errors.

Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.