After some head-scratching, I think I was able to make sense of the "transformation" that the author uses in equation 5.4 to yield the correct expectation for $v_{\pi}(s)$. I've introduced some notation to help clarify what's going on. In particular, let $G_t^{(\pi)}$ denote the return after time step $t$ by following policy $\pi$. Then our goal is to estimate $v_{\pi}(s)$ for the target policy $\pi$ from an episode generated under behaviour policy $b$. Note that the state-value function for $b$ is defined as
$$v_b(s) = E[G_t^{(b)} | S_t = s]$$
We now apply the multiplication by the importance-sampling ratio proposed in equation 5.4 and expand using the definition of the expected value of $G_t^{(b)}$ as follows.
$$E[\rho_{t:T-1}G_t^{(b)} | S_t = s] = \sum_{g \in supp(G_t^{(b)})} \rho_{t:T-1} \cdot p_{G_t^{(b)}}(g) \cdot g$$
However, note that the probability of a particular return occurring under $b$ is just the probability of its sequence of state-action transitions (called the "trajectory" of a return in the text) occurring under $b$. Thus $p_{G_t^{(b)}}(g)$ is simply the probability of a particular trajectory
occurring under $b$, as given in the following excerpt from the text for the example of an arbitrary policy $\pi$.
Recall $p$ here is the state-transition probability. We now substitute this expression for $p_{G_t^{(b)}}(g)$ into our expanded version of equation 5.4, and substitute the definition of $\rho_{t:T-1}$ to obtain:
$$E[\rho_{t:T-1}G_t^{(b)} | S_t = s] = \sum_{g \in supp(G_t^{(b)})} \left(\frac{\prod_{k=t}^{T-1}\pi(A_k|S_k) \cdot p(S_{k+1} | S_k, A_k)}{\prod_{k=t}^{T-1}b(A_k|S_k) \cdot p(S_{k+1} | S_k, A_k)} \cdot \prod_{k=t}^{T-1}[b(A_k|S_k) \cdot p(S_{k+1} | S_k, A_k)] \cdot g \right) $$
Canceling like terms gives us
$$= \sum_{g \in supp(G_t^{(b)})} \prod_{k=t}^{T-1}[\pi(A_k|S_k) \cdot p(S_{k+1} | S_k, A_k)] \cdot g$$
Note that, as discussed earlier, $\prod_{k=t}^{T-1}[\pi(A_k|S_k) \cdot p(S_{k+1} | S_k, A_k)]$ is the probability of a given return trajectory occurring under the target policy $\pi$. In other words, we have:
$$ = \sum_{g \in supp(G_t^{(b)})} p_{G_t^{(\pi)}}(g) \cdot g$$
The support of $G_t^{(\pi)}$ is the same as the support of $G_t^{(b)}$: the support of either random variable is the set of all state-action sequences (trajectories) possible after a particular time step t. We conclude that
$$E[\rho_{t:T-1}G_t^{(b)} | S_t = s] = \sum_{g \in supp(G_t^{(\pi)})} p_{G_t^{(\pi)}}(g) \cdot g = E[G_t^{(\pi)} | S_t = s] = v_{\pi}(s)$$
