Ah. Finally a topic I know something about !
There are many places in physics where the YB equation pops up. I can think of two at the moment.
a. Exactly solvable lattice models
b. Quantum Computation (QC)
It is the second application I find most exciting, so I'll focus on it.
The canonical reference (IMHO) on the link between the YB equation and QC is the wonderful paper by Lomonaco and Kauffmann (LK04) http://arxiv.org/abs/quant-ph/0401090
In topological quantum computation, the hope is to be able to perform unitary operations on qubits by moving them around each other. A typical arena is a 2D electron gas, where our qubits are the quasiparticles of the system. In 2D when we exchange two objects we get a richer symmetry group than in 3D, where we get the permutation group whose eigenvalues $ \pm 1$ correspond to the case of bosons and fermions respectively. However, in 2D this symmetry group is enlarged to the braid group - one can exchange two objects by moving them around such that their worldlines "braid" around each other. This braiding cannot be eliminated by deforming the trajectories, because we don't have the third dimension to utilize.
Anyhow to cut a long story short, the YBE can be shown diagramatically as a relationship between three particles under exchange (see fig. 1 on pg. 8 of above ref.). What LK04 then show is that solutions of the YBE are unitary matrices which are universal for quantum computation. In much the same way that any classical binary circuit can be built out of NAND gates along, any quantum circuit can be built out of a set of universal quantum gates.