For some reason, the question seems to be asking for an *algebraic* (number- or ring-theoretical) justification for residual finiteness (and implicitly LERF, though in fact the correct statement is that every hyperbolic group is residually finite if and only if they are all QCERF, as proved by Agol--Groves--Manning).  Recall that residual finiteness means that the trivial subgroup is separable, and QCERF means that all quasiconvex subgroups are separable.

But the principal motivation is topological (as Benjamin Steinberg has pointed out in comments)---this was why residual finiteness was key to the proof of the Virtual Haken Conjecture.  Precisely, Scott observed that if $\Gamma=\pi_1X$ and $X$ is a cell complex, then a subgroup $H$ is separable if and only if, for any compact subet $K$ of the associated covering space $X^H\to X$, there is a factorization

$X^H\to X'\to X$

where $X'\to X$ is finite-sheeted and $K$ embeds into $X'$.

In other words, QCERF would enable us to promote immersions to embeddings in a finite-sheeted covering space.  As a I said in comments, this would be a huge step forward in our understanding of the topology of such spaces $X$.  There are lots of applications: the paper by Friedl and Vidussi mentioned by Ian Agol in the comments is a nice one.

I'll finish off by buying into the premise of the question for a moment and mentioning a couple of algebraic applications.  One is not directly an application of residual finiteness, but it's in the same circle of ideas, and although the OP makes rather unreasonably narrow demans for motivation, I hope (s)he will allow me this much leeway.

1. Bridson and Grunewald answered a question of Grothendieck by exhibiting a map between a non-isomorphic pair of residually finite, finitely presented groups, which induced an isomorphism on profinite completions.  (As in Francesco's answer, the importance of residual finiteness here was that the \'etale fundamental group is the profinite completion.)  They made direct use of Wise's construction of certain residually finite hyperbolic groups.

2. Let $M$ be a hyperbolic manifold.  By the proof of the Virtual Haken theorem, we know that the first Betti numbers of finite covers of $M$ can be taken as large as we like.  I believe that number theorists would be very excited if, in the case where $M$ is arithmetic, the same result could be proved using the congruence covers of $M$.  Clearly the Virtual Haken theorem, while not enough for them, is a good start!

...

There are others, but this seems like enough for now.