Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an example of a question about $\Gamma$ that was answered by considering these dynamics?
Where to begin!
The ergodicity of non-compact subgroups (singular tori) was used by Margulis to prove that higher rank lattices $\Gamma$ are arithmetic.
Once you have that $\Gamma $ is arithmetic, this has the following consequences: (1) if $Comm (\Gamma)$ is the abstract commensurator, then $Comm (\Gamma)/\Gamma$ is infinite. (2) The cohomology groups $H^*(\Gamma,\mathbb{Z})$ are finitely generated abelian groups (Raghunathan). (3) The Group $\Gamma$ is finitely presented (Borel-Harishchandra).
The Oppenheim conjecture about quadratic forms could be interpreted as a property of the dynamics of $SO(2,1)$ action on $SL(3,\mathbb {R})/SL(3,\mathbb{Z})$ (and was proved by Margulis).
That normal subgroups of torsion free higher rank arithmetic groups $\Gamma $ have finite index is also proved by dynamics of toral actions on $G/\Gamma$ (Margulis).
In Zimmer's book, the proof of Borel density theorem (that a lattice is Zariaski dense in $G$) is proved using dynamics of $G$ action on $G/\Gamma$ and also $G$ action on projective space.
Yet another Margulis theorem says that higher rank arithmetic groups are not free products (or not even amalgams) ; one part of the proof uses ergodicity of actions of singular tori on $G/\Gamma$.
I am sure @YCor knows many more (and also recent) examples.
You may also see https://mathscinet.ams.org/mathscinet-getitem?mr=1898148 for some more examples.
There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem. Actually if you will investigate Ratner's original formulations of her theorems, she usually only assumes that $\Gamma$ is discrete, and then concludes something about it having finite-volume.
Furstenberg's proof of Borel's density theorem also come into mind.
Sort of related to dynamics - Kazhdan's property T (i.e. very strong mixing statement) was developed by Kazhdan (and Margulis) to answer Selberg's conjectures about the abelinzation of a lattice.
A more recent development, using the quantitative equidistribution of semisimple periods (proved by Einsiedler-Margulis-Venkatesh), the authors (with the addition of Mohammadi) managed to show ''transport of spectral gap'' of products allowing one to derive a new proof of property tau (for most types of groups, definitely the weird ones included).
Another example can be seen in the proof of Einsiedler-Lindenstrauss-Michel-Venkatesh of Duke's (Linnik-Skubenko) theorem. The statement there is about the $\Gamma$-action on $G/A$ and then they use duality to transfer it to problem about $A$-action over $\Gamma \backslash G$. [There's also a whole industry of counting problems and Diophantine approximation problems related to lattice actions a-la Duke-Sarnak, see the recent book by Gorodnik-Nevo.]
I believe the question is exactly what you call ''Homogeneous dynamics''. I guess the subject's founding fathers, back in Jerusalem, will say it is a mix of ergodic theory, representation theory (and so harmonic analysis), Lie and algebraic group theory with some inputs from geometric group theory and number theory (both analytic, geometry of numbers and algebraic nt), rather than just focusing on the ergodic theory of the matters. So I guess the question is what you are willing to assume and what are you after exactly. If you just want direct corollaries of the pointwise ergodic theorem, then it is a whole different story.
Kahn and Markovic solved the surface subgroup problem and Ehrenpreis conjecture making use of exponential mixing of the geodesic flow on compact hyperbolic manifolds. The geodesic flow may be thought of as homogeneous dynamics on a diagonal subgroup $H< G$ acting on $G/\Gamma$, although the proof by Cal Moore only refers to the unit tangent bundle (which is where the geodesic flow lives). At least in the case of $PSL_2(\mathbb{R})$ (for hyperbolic surfaces), this is the same thing.
In turn, the surface subgroup problem tells us many more interesting things about $\Gamma$ when $\Gamma$ is a closed hyperbolic 3-manifold group.