You may not see this answer, but as an individual memo, I answer your question.
From eq. (4.18), you can get
$$ H_I(t) = \left( i \frac {\partial} {\partial t} U(t,t_0) \right) U^\dagger(t,t_0). $$
Then, substitute this eq. for the right-hand side of eq. (4.24).
If you do so,
\begin{align*}
&...