What does this equation mean?

Because $H_{SB}$ conserves total excitation number, the dynamics starting from a single excitation stays confined to the two-dimensional subspace spanned by |e,g$\rangle$ (system excited, bath in its ground state) and |g,e$\rangle$ (the reverse). On resonance, both basis states carry the same bare energy, +$\hbar$$\omega$/2 - $\hbar$$\omega$/2 = 0 , so $H_f$ restricted to this subspace is proportional to the swap operator between the two, and the Schrödinger equation solves in closed form. Starting from |$\psi(0)$$\rangle$ = |e,g$\rangle$ , $|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle$. the standard two-level exchange solution, exact for all t under this Hamiltonian. From it, $\langle$ $H_S$ $\rangle(t)$ = ($\hbar$$\omega$/2)[$\cos^2(gt)$ - $\sin^2(gt)$] = ($\hbar$$\omega$/2)$\cos(2gt)$ , while $\langle$ $H_f$ $\rangle$ is exactly time-independent — a general fact for any state evolving under its own generator, not special to this one — and equal here to $\langle$ e,g|$H_f$|e,g$\rangle$ = $\hbar$$\omega$/2 - $\hbar$$\omega$/2 + 0 = 0 , since $H_{SB}$ is purely off-diagonal in this basis. The ledger reading follows immediately: