You are right. Theorem 3.4 shows that the meromorphic functions in the extended complex plane agree with the rational functions.
Moreover it is clear that a rational function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles.
Hence the same holds for a meromorphic function, but this was not clear before Theorem 3.4 was proved.
The statement after "as a consequence" is certainly a little inattentiveness of the authors.
Finally note that it is essential to consider meromorphic functions in the extended complex plane. For a meromorphic functions in the complex plane the assertion of Theorem 3.4. is false. A rational function in the complex plane is of course meromorphic and it is moreover determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles. However, the are more meromorphic functions than rational functions. As an example take the functions $\sin z$ and $e^z \sin z$ which are entire (hence meromorphic) but are not rational. Both have the same set of zeros with multplicity $1$ and no poles.
The problem is that each meromorphic function in the extended complex plane restricts to a meromorphic function in the complex plane, but not each meromorphic function in the complex plane is obtained as such a restriction. In fact, if $f$ is meromorphic the complex plane, then $\infty$ may be an essential singularity $f$ or it may be a cluster point of zeros or poles of $f$. Both situations prevent to consider $f$ as a meromorphic function in the extended complex plane.