Some topological solitons formed from bosonic fields have fermionic statistics. Why?
A short answer: The soliton in bosonic field theory can be fermionic because the model secretly contains massive fermions at high energies.
This is because in order to define an bosonic field theory, we need to non-perturbatively regulate the field theory. So let us put the bosonic field theory on a lattice to non-perturbatively regulate the theory (otherwise, the field theory is not even well defined at non-perturbative level). The claim is that to produce the needed topological term that makes the soliton an fermion, the lattice model must contain fermions with finite energy gap.
A more precise claim: Any gapped bosonic model on lattice that have unique ground state on closed space of any topology do not contain fermionic quasiparticles.
Basically, "gapped bosonic model on lattice that have unique ground state on closed space" implies that the model has no topological order. The only way to have emergent fermion from a bosonic lattice model is to have a non-trivial topological order. Also see a related discussion, where I claim that, in lattice bosonic model, the emergent fermion must appear together with emergent gauge theory at low energies. Skyrme-model contain no low energy gauge theory. This is why I claim that the Skyrme-model secretly contains massive fermions at high energies.
There are many examples. Moshe focused on lower-dimensional theories; for example, in 1+1 dimensions, bosonic and fermionic conformal field theories are typically totally equivalent to each other (and one may get fermions as kinks of the bosons, and bosons as bilinear currents of the fermions). I will focus on higher-dimensional theories.
I don't really know how to get fermions in the $D=26$ bosonic string theory. But in other, $D=10$ string theories that normally contain bosons only, fermions may be obtained - see e.g. this paper by Justin David, Shiraz Minwalla, and Carlos Nunez (whose writing I remember pretty well, back from Santa Barbara 2001 etc.)
The basic trick, known already in 3+1-dimensional bosonic gauge theory (by Jackiw-Rebbi-Hasenfratz-’t Hooft), is "spin from isospin".
The soliton is only invariant under the "diagonal group" of the rotations and some internal isospin group. Noether's theorem dictates that the right definition of the spin of the excitations is associated with whatever symmetry we have. In this case we have the diagonal symmetry, so the new conserved angular momentum will be the diagonal one. Because half-integer-isospin excitations are possible, they will look like half-integer-spin excitations of the soliton.
The fermionic statistics has to do with a minus sign acquired when two objects are interchanged. Usually this is unambiguously attributed to statistics, as opposed to interaction between the objects, since any interaction falls off rapidly when they are far apart, but the relative phase acquired due to interchange is the same whether they are close or far.
However, in low dimensional systems the interaction strength between well-separated objects does not always fall off very fast (or not at all). So, one can get fermionic (or more generally anyonic) behaviour from interaction. For some solitons the interactions is such that they acquire a phase when interchanged, no matter how far apart they are.
This arguments only shows it can happen in low dimensional systems, more detailed arguments will show you how it happens (for example google sine-Gordon and Thirring models, or bosonization). The why questions I am never sure how to answer.