Converting generalized NFAs to NFAs
You are correct in your understanding of the construction: For every transition arrow, say from state $A$ to state $B$ in the GNFA labeled by a regular expression $R$, you replace that transition by the NFA, $N$, for $R$, with an $\epsilon$-transition from $A$ to the start state of $N$ and $\epsilon$-transitions from the final states of $N$ (which you then "un-finalize") to state $B$. Do this for every transition in the GNFA and you'll wind up with an ordinary NFA. 

To show that the two accept the same language, it's enough to rely on the previous result (Lemma 1.55 in Sipser) that any regular expression denotes a language that is recognized by a NFA.