It might be worth pointing out that topological chiral homology is a specialization to the topological (or locally constant) setting of a construction that originated (AFAIK) in conformal field theory, namely the functor of conformal blocks for a vertex algebra and its derived formulation
as chiral homology, due to Beilinson and Drinfeld.
As such there are loads of concrete applications and calculations.

Perhaps the best known is the proof of the Verlinde formula, calculating the dimension of spaces of nonabelian theta functions on a Riemann surface by realizing them as conformal blocks and using the gluing properties of conformal field theory to reduce to a calculation in the representation theory of loop groups. (Recently Gaitsgory gave a derived enhancement of this calculation,
showing that the full chiral homology of the integrable level k vertex algebra of a loop group is still the space of nonabelian theta functions).

Perhaps my favorite application still is the one that motivated Beilinson and Drinfeld to introduce chiral homology in the first place. Namely, they wished to construct special $D$-modules on the moduli of $G$-bundles on a Riemann surface which were eigen-objects for Hecke functors, with eigenvalues particular $G^\vee$-local systems on the surface. (This is the central goal of the geometric Langlands program). They achieved this goal when the local system is a so-called "oper" by showing first how to use the Feigin-Frenkel theorem (a loop algebra version of the Harish Chandra isomorphism for the center of the enveloping algebra) to solve a local version of the problem.
In other words, they showed that Feigin-Frenkel provide representations of the loop algebra which are eigenobjects for Hecke functors. But then the mechanism of conformal blocks (or chiral homology) allowed them to globalize this construction on a Riemann surface, resulting in the desired global objects. The in-depth examination that resulted of what vertex algebras really mean geometrically led to the notion of factorization algebras and chiral homology which are now becoming so influential, thanks to work of (alphabetically) Costello, Francis, Gaitsgory, Gwilliam, Lurie, Rozenblyum and others.

The Beilinson-Drinfeld idea of chiral homology captures in a profound geometric way the adelic structure of moduli spaces of bundles on curves. The recent work of Gaitsgory and Lurie (as well as Rozenblyum, Barlev, Zhu and others) greatly furthers the Beilinson-Drinfeld dream of realizing the entire geometric Langlands correspondence on a Riemann surface as the "integral" (chiral homology) over the surface of a simpler local statement.

In short, chiral homology is providing a formula for the "path integral" over a Riemann surface, telling us how to integrate more categorical but simpler local invariants to get global invariants.
For an introduction to more recent developments (and in particular the migration of factorization algerbas outside of the confines of CFT or TFT), I recommend Costello's ICM address (about recovering the Witten genus through factorization algebras).
Also in a sense made precise I think by Ayala and Rozenblyum,
all the values of any extended TFT (in the sense of the cobordism hypothesis) are given by calculating chiral homology of an appropriate local quantity (i.e., a higher category is a kind of "partially defined E_n algebra"), so in principle the examples are everywhere!