There is a very nice hydrodynamic interpretation, going back to Riemann and Klein, and popularized by Courant (Hurwitz and Courant, Function theory). For those who do not read German, there is a modern exposition in French: E. Ghys and D. Smai, Six lecons autour des surfaces de Riemann
Edit. Klein's book is "On Riemann's theory of algebraic functions", the only one of the three which has been translated into English.