At the start of section 2 of this article, the authors say the following (emphasis mine):
The work of Bernard Riemann in the 1850s and 1860s changed the direction of the theory of algebraic curves, replacing curves in the complex projective plane with branched coverings of the complex projective line—the Riemann sphere—and questions about tangents and secants by questions about spaces of meromorphic functions with bounded pole orders.
The italicised line looks unfamiliar to me.
What is the relation between tangents and secants of plane curves and meromorphic functions with bounded pole orders?
I understand Riemann surface theory, covering spaces, basics of algebraic curves and complex analysis.