You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the classical case (for me, anyway).

As setup, we have a Hilbert space of states $V$ and a state vector $\psi \in V$ which evolves according to a Hamiltonian $H$, meaning (in natural units, so $\hbar = 1$) that

$$\psi(t) = e^{iHt} \psi.$$

Suppose in addition that we have a one-parameter family $g(t)$ of symmetries of our quantum system, meaning that $g(t)$ commutes with the Hamiltonian: $g(t) H = H g(t)$. In particular, $g(t)$ is a one-parameter family of unitary maps, and so by Stone's theorem $g(t)$ must have the form

$$g(t) = e^{iAt}$$

for some self-adjoint operator $A$ (which in physically relevant examples is often unbounded). (In the finite-dimensional case this is just saying that the Lie algebra of the unitary group is the Lie algebra of skew-adjoint matrices.) **Noether's theorem** is the observation that this means $A$ must also commute with $H$, which means that it is an observable which is conserved under time evolution in the sense that

$$e^{iHt} A e^{-iHt} = A$$

(time evolution of $A$ looks like conjugation in the Heisenberg picture). This general observation reproduces many of the familiar conserved quantities in physics. To give two examples:

- If $g(t)$ is translation in a space direction, $A$ is momentum in that direction. For example, if $g(t)$ is translation in the $x$ direction, $A = i \frac{\partial}{\partial x}$.
- If $g(t)$ is rotation around an axis, $A$ is angular momentum around that axis.

Because $A$ is a conserved quantity, it's natural to break up $V$ into eigenspaces of $A$ (corresponding to states where $A$ has a definite value), and the reason is that time evolution preserves all of these eigenspaces. This means that the statement "$\psi$ belongs to such-and-such eigenspace" is physically meaningful, e.g. the statement that $\psi$ has a fixed momentum.

The connection to representation theory comes from thinking of $g(t)$ as a representation of $\mathbb{R}$, so that the eigenspaces of $A$ are the isotypic components of this representation. Irreducible representations correspond to eigenvectors, which are, as above, states where $A$ has a definite and fixed value.

Now many physical systems come with a noncommutative group of symmetries, so it's natural to generalize $g(t)$ to an action of a nonabelian Lie group $G$, for example $SO(3)$, which we again posit to commute with $H$. What we might call the **generalized Noether theorem** is the observation that this implies that time evolution preserves the decomposition of $V$ into isotypic components of this representation, so it's again physically meaningful to say things like "$\psi$ belongs to the isotypic component corresponding to such-and-such irreducible representation" (in physics language, "$\psi$ transforms under...") because such statements are preserved by time evolution. This is the beginning of Wigner's classification (although that classification is relativistic whereas this story I've been telling is decidedly not so some tweaks need to be made). So you can think of the irrep a state belongs to as a "generalized conserved quantity."

(The reason we want to consider irreps is that they give more precise information while continuing to be physically meaningful. I could talk about e.g. particles whose momentum lies in a certain range instead of talking about particles with particular values of their momentum, but the latter is more precise so I do that first.)

The relationship to the groups $U(1), SU(2), SU(3)$ appearing in the standard model requires a bit more elaboration, because these groups don't act by physical symmetries (like the Poincare group) but by gauge symmetries. But that's a story that's a bit outside my competence to describe the physical relevance of. I can tell you that the $U(1)$ factor corresponds to charge conservation.

I should mention that I asked exactly this question awhile ago, and after thinking about the answer I got I wrote this blog post about a toy model of quantum mechanics on a finite graph that you might find helpful.