The logo has a piece of the complex upper half plane divided into fundamental domains for the action of $SL_2(\mathbb{Z})$ by Möbius transformations (which are hyperbolic isometries - see the appropriate entry in McMullen's gallery or Wikipedia). There are no Ford circles in sight, but you may have been confused by the semicircular regions on the bottom. Those regions are unions of fundamental domains, and are cut out by geodesics in the $SL_2(\mathbb{Z})$ orbit of $1/2 \leftrightarrow \infty$ (which forms a part of the boundary of a few fundamental domain)domains). Ford circles come from the $SL_2(\mathbb{Z})$ orbit of the horocycle $\operatorname{Im}(\tau) = 1$, and horocycles have constant nonzero geodesic curvature (imagine driving with your steering wheel turned a bit to the right, but never returning to where you started).
The generating function for $\tau$ is the 24th power of Dedekind's $\eta$ function (often written as the discriminant form $\Delta$), and it is a function on the upper half plane that is invariant under the weight 12 action of $SL_2(\mathbb{Z})$. Up to normalization, it is the unique lowest weight nonzero level 1 cusp form, and this makes it automatically a Hecke eigenform. Both Mordell's proof of the earlier Ramanujan conjectures and Deligne's proof of the growth conjecture use this fact in an essential way. I should note that Deligne proved the Ramanujan conjecture as a corollary of the Weil conjectures, not "in the process".
The connection between hyperbolic geometry and the Ramanujan conjecture is not particularly strong, as far as I know (but I would be happy to be shown the errors of my ways).
