How can a benign tumor of the middle ear lead to infection in the brain?

Obstruction. One possibility is due to "mass effect" where the tumor blocks normal drainage patterns and leads to accumulation of fluids which get infected. The middle ear is right next to the brain so things can spread directly, or sometimes erode through the bone.
Cholesteatoma. This is likely a cholesteatoma which can erode the skull base and cause infection in the brain... This requires surgical removal.