If I remember correctly, isomorphism classes of simple objects correspond to different types of particles (which is assumed to be finite), furthermore more is structure is usually needed than a fusion category, for example braiding (which is the reason why anyons are so interesting). Let me be very concrete. A physically (and experimentally) relevant category has three isomorphism classes of simple objects $(\mathbf 1, \psi, \sigma)$ with the non-trivial fusion rules

$$ \psi\otimes\psi = \mathbf 1, \quad \psi\otimes\sigma =\sigma\quad \text{and}\quad\sigma\otimes\sigma = \mathbf 1 \oplus \psi,$$
where $\sigma$ is the so-called Ising anyon (and $\mathbf 1$ is the unit object). These quasi-particles are conjectured to show up in the $\nu = 5/2$ plateau in fractional quantum Hall systems and in $p+ip$ wave superconductors.

These fusion rules can be used to construct the ground state Hilbert space, which are given through the space of morphisms between simple objects. Defining $V_{ab}^c = \text{Hom}(a\otimes b,c)$, the Hilbert space for two Ising anyons is $V_2 = V_{\sigma\sigma}^{\mathbf 1}\oplus V_{\sigma\sigma}^{\psi}$ which is two-dimensional. For $2n$ anyons, the ground state is $\text{dim}V_{2n}= \text{dim}V_{2n\sigma}^{\mathbf 1} + \text{dim}V_{2n\sigma}^{\psi} = 2^{n-1}+2^{n-1} = 2^n$ dimensional (this is nicely seen using a graphical notation for morphisms, se references below). Using the fusion rule $\sigma\otimes\sigma = \mathbf 1 \oplus \psi$ one can solve the pentagon and hexagon equations for the $F$ and $R$ symbols, which when combined gives rise to a representation of the Braid group $B_{2n}$ (more preciely, the mapping class group of the n-punctured sphere=braids group + Dehn twists) on the ground state Hilbert space$V_{2n}$.

Thus one physical consequence of these direct sums is that the ground state is degenerate and the anyons have highly non-trivial statistics, the ground state wave function transforms under (higher dimensional) representation of the braid group when the particles are adiabatically moved around each other. This property of (non-abelian) anyons has given rise to the idea of using them for quantum computation (another property is their non-local nature, which partially saves them from decoherence).

To get a more physical idea of what fusion (or collision as you call it) of particles mean, one can look at the concrete $p+ip$ wave superconductors. In such superconductors zero (majorana) modes can be bound to the core of Abrikosov vortices, where for 2n vortices there will be $2^n = 2^{n-1} + 2^{n-1}$ fermionic states. This means that it takes two majorana fermions to get one conventional fermion. When the vortices are spatially separated, the state in the core of the vortex cannot be measured by local measurements. In the above notation; $\sigma$ is a vortex, $\psi$ an electron, and $\mathbf 1$ a cooper pair ("the trivial particle"). With this identification, the fusion rules say that fusing two electrons ($\psi\otimes\psi = \mathbf 1$) gives a cooper pair which vanishes in the condensates, while fusing two vortices ($\sigma\otimes\sigma = \mathbf 1\oplus\psi$) give either nothing **or** an electron.

Thus the physical meaning of these direct sums of simple objects has something to do with the possible outcomes when we measure the state after fusing two particles. In this way (non-abelian) anyons can be used to construct qubits, by braiding them one can do a computation, in the end one can fuse them and measure the resulting state.

**References:**
You can read appendix B and then chapter four in this thesis, to get a more precise description of how braided ribbon categories and anyons are connected. These lecture notes by John Preskill gives a more physical insight, in section "9.12 Anyon models generalized" category theory is used to formulate the physics (although category theory language is not used, and might be annoying if you are a mathematician). For a mathematician a better reference is

Last but not least, the canonical reference for non-abelian anyons is the review paper

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