Groups are tied intimately with symmetries. I looked up once what the connection between the was, and the results were hieroglyphics. But they were useful hieroglyphics... after a lot of wikidiving into category theory!
At the highest level, groups are mathematical objects which describe how symmetries compose. Often just seeing how they compose provides great insight into the possibilities that can occur.
For example, one of the classic rules in physics is time-symmetry. This means that you may not know what's going to happen at t=1, but you're pretty sure that if you started the experiment 10 seconds later, whatever would have happens at t=1 happens at t=11. This behavior is associated with an $R$ group. The similar behavior in 3-space is associated with an $R^3$ group, so a property which is the same at any time or place exhibits $R\times R^3$ symmetry.
The SU class of groups is associated with $n\times n$ unitary matrix multiplication, so things with SU(2) symmetry behave the same when multiplied at the right place by a 2x2 unitary matrix.
What's great about this is that it empowers all of the other math we might do later. It lets us ignore any possible answer which violates said symmetry. And, as it turns out, much of that can be done in group-theory land without actually having to operate on the real equations. We can often predict some behaviors, such as the existence of half spin solutions, simply from what must be true to keep that symmetry functioning.
Symmetries are a powerful thing. There is a famous derivation by Emmy Nother which proved that, in Lagrangian mechanics, every continuous symmetry yields a conserved value. Time symmetry, as mentioned above, is actually associated with conservation of energy. Spatial symmetries result in conservation of momentum. And in the quantum world, the SU(3) symmetry found in the equations for color charge yield conservation of color. Group theory is simply the power-tool used to leverage these symmetries to yield results!
As to your specific example, consider whether you can have a value represented by thirds. You could work the math and prove that such a value will result in a Lorenz symmetry failure. But what about quarters? Nope. Breaks the symmetry. What about irrational values? We could be here all night. But a little trip into group theory and lie algebras immediately tells us that integers and half integers are all we need to consider. Any other solution will not exhibit Lorenz symmetry.
A quick edit (which I may properly incorporate later).
In category theory, a symmetry of some function $\phi$ is a pair of functors $(\alpha, \beta)$ such that $\alpha\circ\phi = \phi\circ\beta$. The class of all symmetries for some $\phi$ forms a group (it has the properties that a group hash). Now that's basically gibberish if you ask me, but it is meaningful. Let's put it in function notation: $\alpha(\phi(X))$ = $\phi(\beta(X))$. It's a bit easier to see the power. Now look at the right side. I'm modifying X, the input to $\phi$ with some $\beta$ function. On the right hand side, I'm modifying $\phi(X)$, the output of $\phi$ with some $\alpha$ function.
Now look how powerful that is. This says that if $(\alpha, \beta)$ is in the symmetry group of $\phi$, and I solve $\phi(X)$ once, I can use $\alpha$ to modifying the output to account for any $\beta$ in that group. I've moved the modification from the input side of the process to the output side. To answer your question in the comments, it's not always linear, but it is defined by the symmetry group.
Group theory lets you make statements based on the existence of these symmetric pairs without having to actually identify the functions. Thus you can either take the time to prove that a particular system with a Lorentz symmetry can be represented with integers and half integers (and that you can't simplify it further), or you can just say "All systems with Lorentz symmetry have an associated SO(3,1) group, and SO(3,1) groups can always be represented by integers and half integers, so this system must be representable in the same way.". Now you may come across a far more complex system, with far more details to work with. But if you look at it from a symmetry perspective, and find that the system has a symmetry group of U(1)SO(3,1)SU(2), then you know it has all the charactaristics of any system with SO(3,1) symmetry, without having to run any extra math.