Absolute convergence of $\displaystyle \prod_{k=1}^{\infty} \left( 1+a_k\right)$ means that $\displaystyle \sum_{k=1}^{\infty} \lvert a_k \rvert$ converges.
Further if $a_k$'s are positive, then we have the following inequalities. $1 + \sum_{k=1}^{\infty} a_k \leq \displaystyle \prod_{k=1}^{\infty} \left( 1+a_k\right) \leq \exp \left( \sum_{k=1}^{\infty} a_k \right)$ Hence, if $a_k$'s are positive, then $\displaystyle \prod_{k=1}^{\infty} \left( 1+a_k\right)$ converges iff $\displaystyle \sum_{k=1}^{\infty} a_k$ converges.
Since we are given that $\displaystyle \prod_{k=1}^{\infty} \left(\dfrac{p_k}{p_k-1} \right) = \prod_{k=1}^{\infty} \left(1 + \dfrac1{p_k-1} \right)$ diverges, we have that $\sum_{k=1}^{\infty} \dfrac1{p_k-1}$ diverges i.e. if $b_n = \displaystyle \sum_{k=1}^{n} \dfrac1{p_k-1}$, then $\displaystyle \lim_{n \rightarrow \infty} b_n = \infty$.
Now note that \begin{align} a_n & = \dfrac1{p_1} + \dfrac1{p_2} + \dfrac1{p_3} + \cdots + \dfrac1{p_n} > \dfrac1{p_2-1} + \dfrac1{p_3-1} + \dfrac1{p_4-1} + \cdots + \dfrac1{p_n-1} + \dfrac1{p_{n+1}-1}\\ & = b_{n+1} - \dfrac1{p_1-1} = b_{n+1} - 1 \end{align} The above is true since successive prime differ at-least by $1$ i.e. $p_k < p_{k+1}-1$.
Now you letting $n \to \infty$, we get what you want.