There is a lot of background here, and again, the best way to learn it is to find a good textbook on the representation theory of finite groups. But here's a brief sketch.
Any finite group $G$ has a finite list of irreducible representations. Maschke's theorem guarantees that any (say, complex, finite-dimensional) representation decomposes into a direct sum of irreducible representations. More concretely, this says that given any representation, we can simultaneously block-diagonalize all of the elements of $G$ so that the blocks correspond to irreducible representations.
$S_2$ has a $2$-dimensional representation $V$ given by its action as permutation matrices. This representation decomposes into a direct sum $V_0 \oplus V_1$ of the trivial representation and the nontrivial (sign) representation. We can say that the elements of $V_0$ transform under the trivial representation, and the elements of $V_1$ transform under the sign representation. (This seems to be physics / chemistry terminology. In mathematics we would just say that $V_0$ is, or perhaps is isomorphic to, the trivial representation, and so forth.)
Okay, so how do we find this decomposition? Write $V = \text{span}(a, b)$ where the nontrivial element $g$ of $S_2$ exchanges $a$ and $b$. Then $g$ fixes the vector $a + b$, so $a + b$ spans the trivial subrepresentation. Next, we want to find a vector that $g$ multiplies by $-1$, and such a vector is given by $a - b$. So $V$ decomposes into a direct sum $\text{span}(a + b) \oplus \text{span}(a - b)$ where $S_2$ acts by the trivial representation on the first summand and by the sign representation on the second summand.
There is a more general projection formula here, but to learn what it is, again, you really should find a good textbook on the representation theory of finite groups.
Edit: I have skimmed through http://www-users.math.umn.edu/~mille003/Lie_theory_special_functions_paper.pdf, by Willard Miller, referred to by @rrogers in the comments. In that paper, the author mentions that a class of special functions occur as matrix elements of irreducible representations: see the Peter-Weyl theorem (https://en.wikipedia.org/wiki/Peter–Weyl_theorem) and the Schur orthogonality relations (https://en.wikipedia.org/wiki/Schur_orthogonality_relations). As an example, if $G = SO(3)$, then the spherical harmonics are related to the matrix elements, when $SO(3)$ is described using Euler angles.
The author also mentions other ways in which symmetry/group theory and special functions are related (I quote: as basis functions for Lie group representations, as solutions of Laplace-Beltrami eigenvalue problems with potential via separation of variables, as Clebsh-Gordan coefficients).
It therefore seems that special functions are related to Lie groups in many different ways.
I am not well versed in special functions, but the way I think of them, which may not apply to all cases in which the functions are called special, is as special bases elements of the space of invariants of a system of differential equations (usually coming from physics), that are associated to the group of symmetries of that system.
Old answer: Finally, I can tell you a little about the geometry of the Lie groups you have mentioned, $SL(2,\mathbb{R})$, $SU(2)$ and $SO(3)$. Let me mention Helgason's books which may be useful to you too.
First, these $3$ groups are all "real forms" of the Lie group $SL(2,\mathbb{C})$. $SL(2,\mathbb{R})$ is non-compact, corresponding to a specific "real slice" of $\mathfrak{sl}(2,\mathbb{C})$ (the Lie algebra of $SL(2,\mathbb{C})$), while $SU(2)$ and $SO(3)$ are both compact, and correspond to another "real slice" of $\mathfrak{sl}(2,\mathbb{C})$; however, and this is important to spin geometry, Dirac equation and all that, $SU(2)$ is simply connected, while $SO(3)$ has fundamental group $\mathbb{Z}/(2)$. Topologically, $SU(2)$ is the 3-dimensional sphere $S^3$, while $SO(3)$ is the real projective $3$-space $\mathbb{R}P^3$, which can be thought of as $S^3$ with antipodal points identified.
Now we come to the geometric interpretation of these groups, so to speak. $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm Id\}$ is actually the group of oriented isometries of the hyperbolic plane. If you are familiar with the theory of compact Riemann surfaces, those of genus $g \geq 2$ are called hyperbolic. They can be described nicely as quotients of the hyperbolic plane by a discrete group action. I also think that hypergeometric functions can be interpreted as holomorphic (or meromorphic?) functions on a compact hyperbolic Riemann surface (Check with an algebraic geometer, or online, to get details on the genus $g$ etc.). Thus the link between $SL(2,\mathbb{R})$ and hypergeometric functions is via the hyperbolic plane. The details are unfortunately scattered in the literature, though well known to experts.
Regarding $SU(2)$, it acts naturally on the space of Weyl spinors $S = \mathbb{C}^2$ (by just matrix multiplication). But then $S \odot S$, the symmetric square of $S$, is a complex $3$-dimensional space, on which $SU(2)$ acts via the induced representation. You can check that there is actually a real slice $V \subset S \odot S$ on which $SU(2)$ acts, which actually factors through $SU(2)/\{\pm Id\}$, and which preserves a real inner product (the inner product is induced by a symplectic form on $S$), as well as a choice of orientation of $V$. Thus one gets a group homomorphism from $SU(2)/\{\pm Id \}$ into $SO(3)$, and one can check that it is actually an isomorphism.
Morale: $SU(2)$ is a double cover of $SO(3)$. The former acts on spinors, while the latter acts on vectors. Thus $SU(2)$ is the spin group $Spin(3)$ in dimension $3$.
I hope this helps!
Old Edit: As a comment, it is clear that any (linear) representation of $SO(3)$ is also a representation of $SU(2)$, but the converse is not true. There are representations of $SU(2)$, such as $S$, which are not representations of $SO(3)$. For instance, when $n$ is odd, the $n$-th symmetric square of $S$ is an irreducible representation of $SU(2)$, which is not a representation of $SO(3)$. These representations correspond to spins that are half an odd integer ($1/2$, $3/2$, $5/2$ etc), and are thus associated to fermions, rather than bosons.
Best Answer
Let $G$ be the circle, thought of as complex numbers of norm one. The Hilbert space $L^2(G)$ of complex valued functions on $G$ carries a natural unitary representation of $G$ by the rule $$ g \cdot f(x) = f(g^{-1}x). $$ But $G$ is compact, so representations are (Hilbert space) direct sums of irreducible representations, and $G$ is abelian, so its irreducible representations are one-dimensional -- in this case, they are in bijection with $\mathbb{Z}$, the map being $z \mapsto z^n$. Let's call this representation $\mathbb{C}(n)$.
So we have $$ L^2(G) = \widehat{\bigoplus_{n \in \mathbb{Z}}} \mathbb{C}(n). $$
So we can write functions on $G$ as convergent sums of functions of the form $z \mapsto z^n$, which is to say, we can write periodic functions on $\mathbb{R}$ with period one as convergent sums of functions $x \mapsto \exp(2 \pi i n x)$, which is of course the classical Fourier series.
Similar story with $\mathbb{R}$ and the Fourier transform, except you can't hope for something as nice as breaking a function into discrete pieces because the representations of $\mathbb{R}$ are indexed by the continuous set $\mathbb{R}$ . You instead get that $L^2(\mathbb{R})$ is a "direct integral" of one-dimensional spaces, which really is just language for stating the Fourier inversion formula.