There are many things going on, some of which depend on the exact variety of gridded ion thruster. I'm referencing chapter 6 of Spacecraft Systems Engineering 4th Edition by Fortescue, Swinerd, and Stark. It's a very affordable textbook which gives a good overview of just about any common system found on unmanned spacecraft. It is not a textbook on electric propulsion.
The first thing to realize is that the ionized propellant is electrically conductive. This is true for the same reason that ion-bearing liquids are electrically conductive. This details of this conductivity are not important to my answer. http://onlinelibrary.wiley.com/doi/10.1002/ctpp.19840240304/abstract
The conductivity of the ionized propellant allows current to flow from the negative grid to the positive grid. This is not the whole story though. The propellant becomes ionized within the chamber, and the resulting free electrons are attracted to the positive grid, which creates a source of inflowing current. These electrons need to be continually removed to keep the grid positively charged. There are also electrons which are carried from the negative grid into space by the propellant.
The complete story of how the ions move around is more complicated, but I think this answers your question.
In summary, the current flow is not a requirement, it's a consequence.