$$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$$
Basic idea: the rotating frame "unwinds" part of the evolution of the quantum state so that the remaining part has a simpler time dependence.
The interaction picture is a special case of the rotating frame.
Consider a Hamiltonian with a "simple" time independent part $H_0$, and a time dependent part $V(t)$:
$$H(t) = H_0 + V(t) \, .$$
Denote the time evolution operator (propagator) of the full Hamiltonian $H(t)$ as $U(t,t_0)$.
In other words, the Schrodinger picture state obeys $\ket{\Psi(t)} = U(t, t_0) \ket{\Psi(t_0)}$.
The time evolution operator from just $H_0$ is (assuming $H_0$ is time independent, or at least commutes with itself at different times)
$$U_0(t, t_0) = \exp\left[ -\frac{i}{\hbar} \int_{t_0}^t dt' \, H_0(t') \right] \, .$$
Note that
$$i\hbar \partial_t U_0(t, t_0) = H_0(t) U_0(t, t_0) \, .$$
Define a new state vector $\ket{\Phi(t)}$ as
$$\ket{\Phi(t)}
\equiv R(t) \ket{\Psi(t)}$$
where $R(t)$ is some "rotation operator".
Now find the time dependence of $\ket{\Phi(t)}$:
\begin{align}
i \hbar \partial_t \ket{\Phi(t)}
=& i \hbar \partial_t \left( R(t) \ket{\Psi(t)} \right) \\
=& i \hbar \partial_t R(t) \ket{\Psi(t)} + R(t) i \hbar \partial_t \ket{\Psi(t)} \\
=& i \hbar \dot{R}(t) \ket{\Psi(t)} + R(t) H(t) \ket{\Psi(t)} \\
=& i \hbar \dot{R}(t) R(t)^\dagger \ket{\Phi(t)} + R(t) H(t) R(t)^\dagger \ket{\Phi(t)} \\
=& \left( i \hbar \dot{R}(t) R(t)^\dagger + R(t) H(t) R(t)^\dagger \right) \ket{\Phi(t)} \, .
\end{align}
Therefore, $\ket{\Phi(t)}$ obeys Schrodinger's equation with a modified Hamiltonian $H'(t)$ defined as
$$H'(t) \equiv i \hbar \dot{R}(t) R(t)^{\dagger} + R(t) H(t) R(t)^\dagger \, . \tag{$\star$}$$
This is the equation of motion in the rotating frame.
Useful choices of $R$ depend on the problem at hand.
Choosing $R(t) \equiv U_0(t, t_0)^\dagger$ has the particularly useful property that the first term in $(\star)$ cancels the $H_0(t)$ part of the second term, leaving
\begin{align}
i \hbar \partial_t \ket{\Phi(t,t_0)}
=
\left( U_0(t, t_0)^\dagger V(t) U_0(t, t_0) \right)\ket{\Phi(t, t_0)} \, .
\end{align}
which is Schrodinger's equation with effective Hamiltonian
$$H'(t) \equiv U_0(t)^\dagger V(t) U_0(t) \, .$$
This is called the interaction picture. It is also known by the name Dirac picture.
