Very roughly, the presence of an R-matrix (a solution of the YBE) provides a mechanism for constructing sufficiently many (infinitely many for field-theoretic models) integrals of motion, and this ensures integrability.
As for the YBE in statistical physics, there is a classical book
R.J. Baxter, Exactly solved models in statistical mechanics.
See e.g. also the references at http://www.encyclopediaofmath.org/index.php/Yang–Baxter_equation