We define a mock theta function as follows:
A mock theta function is a function defined by a $q$-series convergent when $|q|<1$ for which we can calculate asymptotic formulae when $q$ tends to a rational point $e^{2\pi ir/s}$ of the unit circle of the same degree of precision as those furnished for the ordinary $\theta$-function by theory of linear transformation.
I want to understand this definition. Accordingly, if we have a rational point on the unit circle what precisely does happen? Can someone explain.
Since 2008, the number theory community has decided on a bit more "scientific" definition of mock theta function. However, given the context in which you asked the question, I don't assume you want the technical definition of the holomorphic part of a weak Maas form (whatever that means I just need to say this to keep the trolls from accusing me of handwaving).
Just to gather what Ramanujan is thinking, look at THE prototypical theta function when $s=1$ and $r=0$ and $q=e^{-t}$ $$ f(e^{-t}) = \sum_{n=-\infty}^{\infty } e^{-tn^2} = \sqrt{\frac{\pi}{t}}\sum_{n=-\infty}^{\infty } e^{-\frac{n^2}{t}} =\sqrt{\frac{\pi}{t}}f(e^{-\frac{1}{t}}) $$ where you can compute the second identity via poisson summation. This formula is elegant in a two ways.
1) We have an asymptotic formula $f(e^{-t})\approx \sqrt{\frac{\pi }{t}}$ to which the error on this approximation is very very small. The next largest term in the sum is when $n=\pm1$ which for say $t=.1$ is of size $10\sqrt{\pi}e^{-10}\approx 2.5*10^{-5}$ when $t=.01$ this largest term is $6.59*10^{-45}.$ This is what he means by "asymptotic formulae" with "theta precision"
2) The formula is very compact and precise, relating $t\to -\frac{1}{t}.$ In fact,
I will leave it to you to try but you can get a formula for $e^{-t}e^{2\pi i \frac{r}{s}}.$ You will always see $f(e^{-t+ 2\pi i \frac{r}{s}})$ having a really nice formula for estimating $f(q)$ when $t$ is small. If you are more careful you will a pattern or a system for computing these estimates relating to fractional linear transformations $(az+b)/(cz+d)$ if you let $t=\pi i z.$ This is the linear transformations part he is talking about.
Now he lists several objects which behave like theta functions in the same sense we described but they clearly aren't $q$ to a square of sorts so we wouldn't call them theta functions. Hence they are mock theta functions. I will give you one Example in Theorem 2.1 which you probably can see is very complex.