I am sorry to hear that your wife is dealing with fallopian tube cancer, and I hope that her surgery
FIGO stands for "International Federation of Gynecology and Obstetrics" and it is an important international body of gynecology and obstetrics experts who set standards of medical care for many gynecological conditions.
FIGO has produced a cancer staging system for gynecological cancers, which is what you are referring to. This system defines have far the cancer has spread - for example, is it just in the fallopian tube, or has it spread to the ovary, uterus, or even outside of the pelvis. In order to establish the FIGO stage, all of these organs (ovaries, fallopian tubes, uterus) are typically removed and examined for cancer. Biopsies are also taken through the pelvis and elsewhere looking for cancer.
The FIGO stage is very useful, because it defines what therapy will be taken for the cancer going forward. It is a good question to wonder what the use of rating a cancer after removal is, but in truth surgery is the first step for nearly all gynecological cancer. Subsequently, the FIGO stage helps to define what followup chemotherapy
and/or radiation is needed.