I don't understand what $H_{sb}$ physically means or where it comes from.
This, of course, depends on the specific problem at hand - the nature of the bath and the type of interaction. An often considered case is electron-phonon interaction, where an electron is coupled to the (polar) phonon field via something like
$$
H_{int}=-e\mathbf{r}\cdot\mathbf{E}(\mathbf{r}).
$$
Polar phonons correspond to the polarization (i.e., electric field/dipole moment) induced when two atoms in a unit cell of a crystal a displaced from their equilibrium positions. When quantizing the phonon field, the displacement is then treated as the momentum of the oscillator, that is
$$
\mathbf{E}(\mathbf{r})\propto\sum_{\mathbf{k}}\lambda_\mathbf{k}\left(b_\mathbf{k}^\dagger e^{i\mathbf{k}\mathbf{r}} - b_{-\mathbf{k}}e^{-i\mathbf{k}\mathbf{r}}\right)
$$
or something of the kind.
Likewise, if we consider an electron in a harmonic oscillator, its position operator is replace by something like $a^\dagger \pm a$.
There many different types of baths that can be modeled as collections of oscillators (at least in linear approximation) - phonons, plasmons, magnons, etc. Another key point to remark in the coupling is that it is assumed to be linear (dipole approximation), although more complex cases are possible.
Derivations can be found in many solid state and/or many-body texts: Kittel, Mahan, AGD, etc.
Update
Coupling to EM field
Another commonly occurring case is the coupling of electrons (in atoms or solid state) to electromagnetic field. The minimal coupling Hamiltonian is
$$
H=\frac{1}{2m}\left[\mathbf{p}+\frac{e}{c}\mathbf{A}(\mathbf{r},t)\right]^2-e\varphi(\mathbf{r},t).$$
One often chooses to work in Coulomb gauge, where $\nabla\varphi=0$, so that the Hamiltonian becomes
$$
H=\frac{\mathbf{p}^2}{2m}+\frac{e}{2mc}\left[\mathbf{p}\mathbf{A}(\mathbf{r},t)+\mathbf{A}(\mathbf{r},t)\mathbf{p}\right]+\frac{e^2}{2mc^2}\mathbf{A}^2(\mathbf{r},t),$$
The last term is then neglected, as describing two-photon processes that are unlikely for the energies of interest, whereas $\mathbf{A}(\mathbf{r},t)$ can be assumed to be constant on the scales of atoms or the transition regions of interest in semiconductors (the wave length of might is hundreds of nanometers, whereas a size of atom is several angstroms.)
We then have the coupling part
$$
H_{int}\propto \mathbf{p}\sum_\mathbf{k}\left(b_\mathbf{k}+b_\mathbf{k}^\dagger\right)
$$
(See Quantization of EM field.)
Oscillator raising and lowering operators
Note that variables $x$ and $p$ in an oscillator Hamiltonian enter in symmetric way (up to coefficients):
$$
H=\frac{\hat{p}^2}{2m}+\frac{m\omega^2\hat{x}^2}{2}.
$$
E.g., in momentum space $\hat{x}=x,\hat{p}=-i\hbar\partial_x$, and the Schrödinger equation takes form
$$\left[\frac{1}{2m}\frac{d^2}{dx^2}+\frac{m\omega^2x^2}{2}\right]\psi(x)=E\psi(x),$$
with the well-known solutions expressed in terms of Hermit polinomials.
In momentum space we have $\hat{x}=i\hbar\partial_p,\hat{p}=p$ and the SE becomes
$$
\left[\frac{m\omega^2}{2}\frac{d^2}{dp^2}+\frac{p^2}{2m}\right]\phi(p)=E\phi(p),$$
which has the solutions expressed in terms of Hermit polinomials of variable $p$.
Likewise, the definitions of raising and lowering operators are interchangeable in terms of momentum and position: traditionally one defines them in such a way that
$$x\propto a+a^\dagger, p\propto a-a^\dagger,$$
but it could as well be
$$x\propto a-a^\dagger, p\propto a+a^\dagger,$$
the difference being the phase factors of the wave functions.
