As a physicist, I may be able to give a different perspective on this question. In particular, many of the responses so far have been about quantum mechanics and quantum field theory (which involve Lie groups), but if the question is, "Why is the particle content of physics theories derived from Lie groups?" then the answer is not specifically about the theories' quantumness. It's about their geometry, which can be discussed separately from quantum effects.

In 1926, Kaluza and Klein attempted to unify electromagnetism with gravity by proposing a theory of General Relativity with 5 dimensions (4 spatial). Since we don't macroscopically experience this extra degree of freedom, they proposed that it is topologically like a cylinder with a small radius, so small that the extra degree of freedom can't be probed as a direction. This degree of freedom does, however, allow us to encode classical electromagnetism as part of the geometry of space-time. We'll see in a moment that while this formulation isn't exactly right, it does show how the differential geometry concepts of General Relativity can be used in particle physics theories, leading to a unification of all four forces at a classical level. (It's the *quantization* of gravity that's the hard part.)

The Lagrangian of quantum electrodynamics (late 1940's) is just the Lagrangian of the Dirac equation with an additional requirement: that the spinor field $\psi$ has a local $\mathcal{U}(1)$ symmetry. I'll use the same notation as the Wikipedia article, except that I'll use $c = \hbar = 1$. The Dirac Lagrangian

$\mathcal{L_D} = m\bar{\psi}\psi - \frac{i}{2}\left(\bar{\psi} \gamma^\mu (\partial_\mu\psi) - (\partial_\mu\bar{\psi}) \gamma^\mu \psi \right)$ (1)

has a global $\mathcal{U}(1)$ symmetry in that the complex phases of components of $\psi$ cancel in the $\bar{\psi}\psi$ terms: multiplying all instances of $\psi$ by $e^{i\alpha}$ for some constant $\alpha$ would not change the value of $\mathcal{L}$. The Dirac equation does not have a local $\mathcal{U}(1)$ symmetry, that is, invariance under

$\psi(x,t) \to e^{i\theta(x,t)} \psi(x,t)$ where everything is a function of 4-D space-time points $(x,t)$ (2).

If we want to create a new Lagrangian which does have a local $\mathcal{U}(1)$ symmetry, we find that we would need to replace the derivative operators $\partial_\mu$ with

$D_\mu = \partial_\mu - iqA_\mu$ (3)

where $A$ is a new field with the transformation property

$A_\mu(x,t) \to A_\mu(x,t) + \frac{1}{q}\partial_\mu \theta(x,t)$ (4).

The new theory has a Lagrangian

$\mathcal{L_{QED}} = m\bar{\psi}\psi - \frac{i}{2}\left(\bar{\psi} \gamma^\mu (D_\mu\psi) - (D_\mu\bar{\psi}) \gamma^\mu \psi \right) + \frac{1}{4}\left((\partial_\mu A_\nu - \partial_\nu A_\mu)(\partial^\mu A^\nu - \partial^\nu A^\mu)\right)$ (5)

where the last term is required to preserve symmetry under Lorentz boosts (conservation of energy in the new $A$ field). Just following the consequences of a local $\mathcal{U}(1)$ symmetry, we have turned freely-streaming Dirac Lagrangian into the interacting electromagnetic Lagrangian, where we can interpret $\psi$ as charged particle (e.g. electron) waves and $A$ as the vector potential of electromagnetism, which is to say, the photon waves. The transformation of Eqns (2) and (4) is the gauge transformation of electromagnetism: we've learned that the electromagnetic gauge symmetry is fundamentally a local $\mathcal{U}(1)$ symmetry.

Getting back to Kaluza and Klein's theory, a 5th compactified dimension is a little like having a $\mathcal{U}(1)$ invariance at every point in 4-D space, since it's hard to see where we are in the loop of the 5th dimension. It's not exactly the same thing: with an extra dimension, we should in principle be able to perform rotations in which spatial dimensions and the extra dimension mix, while that would not be possible in a 4-D space plus $\mathcal{U}(1)$ fiber bundle. (This difference is perhaps related to the reason Kaluza and Klein's original theory didn't work...?) If we generalize our notion of space-time to include the $\mathcal{U}(1)$ fibers, we can think about electromagnetism and General Relativity in the same terms. For instance, the photon field $A$ plays the same role in the $\mathcal{U}(1)$ symmetry as the connection/covariant derivative in the local Lorentz symmetry of the space-time metric. That is, the classical photon field is the "curvature" of the fiber bundle in the same sense that gravitation is the curvature of space-time.

Moreover, this picture unifying the geometry of electromagnetism with the geometry of gravity also works for all the other known forces. In 1954, Yang and Mills generalized the "local $\mathcal{U}(1)$-to-electromagnetism" idea to work for any Lie group, including non-Abelian ones. The Yang-Mills idea wasn't popular at first because it didn't seem to describe the nuclear strong force (but that was based on a wrong assumption that the nuclear force is a Yukawa interaction). By the late 1960's, Weinberg derived a unified electro-weak theory from local $\mathcal{SU}(2)\times\mathcal{U}(1)$, and Han and Nambu derived a theory of nuclear strong force from $\mathcal{SU}(3)$. (I'm skipping over many important contributions for brevity.) By the mid-1970's or early 1980's, depending on who I ask, this became known as the Standard Model of particle physics because of its experimental success.

We can think about the Standard Model geometrically as an $\mathcal{SU}(3)\times\mathcal{SU}(2)\times\mathcal{U}(1)$ at every point in 4-D space-time, with the gluon, W and Z bosons, and photon being connections through groups at neighboring points of space-time, constantly arranging themselves to hide information about the components of matter fields in all of these "internal" degrees of freedom. The structures of the groups are directly responsible for the charges and interactions of the matter fields (quarks and leptons), but the matter fields themselves are not derived from the groups (supersymmetry might change that part of the picture). There is a direct analogy between these group connections (the gluon, W, Z, and photon) and the space-time connection in General Relativity (which we could call a graviton field, if you wish). I have said nothing at this point about the quantization of all of these fields, which further complicates the picture, especially in the case of gravity!

By the way, I would love to know more about the curvature of fiber bundles, in order to understand the above at a deeper mathematical level. If you have any suggested reading, I'm interested. Thanks!

Group Theory and Physicsfor some history and in particular for how group theory was not really welcome initially and referred to as theGruppenpest. – José Figueroa-O'Farrill Jul 4 '10 at 13:52