= Solution
Write $\omega_z=\gamma B_z$ and use the ordered <spin> basis $(|\uparrow\rangle,|\downarrow\rangle)$. The free <Hamiltonian operator> is $H_0=-\hbar\omega_z\sigma_z/2$, so
$$
V_I(t)=-\frac{\gamma\hbar B_x}{2}\cos(\omega t)
\begin{pmatrix}0&e^{-i\omega_zt}\\e^{i\omega_zt}&0\end{pmatrix}.
$$
Resolving the cosine into its two complex exponentials gives the exact upper off-diagonal factor
$$
\cos(\omega t)e^{-i\omega_zt}
=\frac12\left[e^{i(\omega-\omega_z)t}+e^{-i(\omega+\omega_z)t}\right],
$$
with the complex conjugate in the lower entry. Choose the field orientation with $\omega_z>0$, as implicit in the printed near-resonance condition, and put $\delta=\omega-\omega_z$. The <rotating-wave approximation> discards the terms at $\omega+\omega_z$, retaining the slow terms at $\delta$. The resulting <interaction picture> equations are
$$
\boxed{\frac d{dt}\begin{pmatrix}c_1\\c_2\end{pmatrix}
=\frac{i\gamma B_x}{4}
\begin{pmatrix}0&e^{i\delta t}\\e^{-i\delta t}&0\end{pmatrix}
\begin{pmatrix}c_1\\c_2\end{pmatrix}.}
$$
The factor $1/4$ contains $1/2$ from $S_x=\hbar\sigma_x/2$ and $1/2$ from the cosine decomposition. This is a <Driven two-level Rabi model>, with signed on-resonance coupling $\Omega=\gamma B_x/2$ and generalized <Rabi angular frequency> $\omega_R=\sqrt{\delta^2+\Omega^2}$. Here the <Rabi detuning> uses the paper's convention, drive minus free frequency.
Small detuning alone is insufficient to derive the printed equations. One also requires $|\gamma B_x|,|\delta|\ll\omega+\omega_z$, and an evolution time over which neglected terms stay small. At resonance this is the weak transverse-field regime $|B_x|\ll|B_z|$. The supplied solution is thus an approximate solution for the linearly oscillating field, not an exact one at arbitrary drive strength. If $\gamma B_z<0$, the other exponential is resonant and the physical positive resonance frequency is $|\gamma B_z|$.
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