Regarding the Geostationary Environmental Monitoring Spectrometer (GEMS) itself:
The instrument's purpose is given as
Ozone profile and total-column or gross profile of other species. Tracked species: BrO, NO2, O3, OClO, SO2 and aerosol
That is rather inaccurate and misleading. GEMS does not track BrO or OClO (or ClO). It does track ozone, but the target is tropospheric rather than stratospheric ozone. GEMS is a satellite instrument whose primary goal is tropospheric air quality monitoring rather than stratospheric chemistry.
I think that "Mole fraction" is a relative concentration and "Total column" is absolute?
Mole fraction is the relative concentration of some atmospheric component as a function of altitude. Converting this relative concentration to an absolute quantity (e.g., number of moles, or mass) and integrating from the bottom of the atmosphere to the top yields the "total column" quantity for that component.
Why are atmospheric BrO and ClO important to measure by satellite?
This is the question raised in the title of the question.
A large number of halogenated hydrocarbons indirectly contribute to ozone depletion. Those halogenated hydrocarbons by themselves are not ozone-depleting substances. But if those compounds reach the upper stratosphere, sunlight can split a halogen atom off of those components. Split-off fluorine atoms tend to bind rather quickly with hydrogen and form an extremely stable hydrogen fluoride molecule. Fluorine barely registers as an ozone-depleting substance because hydrogen fluoride is so stable. Stratospheric hydrogen fluoride eventually diffuses into the troposphere, where it gets dissolved by water and falls as rain.
Split-off chlorine and bromine atoms have a different fate. Hydrogen chloride and hydrogen bromide aren't nearly as stable as is hydrogen fluoride. Chlorine and bromine instead alternate between reservoirs such as HCl and hBr and atomic/oxide forms that catalytically deplete ozone. The key reason it is important to monitor these halogens is that most (about 80%) of the chlorine and almost half (40-50%) of the bromine in the stratosphere is anthropogenic.
Molina, Mario J., et al. "Antarctic stratospheric chemistry of chlorine nitrate, hydrogen chloride, and ice: Release of active chlorine." Science 238.4831 (1987): 1253-1257.
Poulet, Gilles, et al. "Role of the BrO+ HO2 reaction in the stratospheric chemistry of bromine." Geophysical research letters 19.23 (1992): 2305-2308.
Choi, Won Jun, et al. "Introducing the geostationary environment monitoring spectrometer." Journal of Applied Remote Sensing 12.4 (2018): 044005.
Kim, Jhoon, et al. "New era of air quality monitoring from space: Geostationary Environment Monitoring Spectrometer (GEMS)." Bulletin of the American Meteorological Society 2019 (2019): 00.
In addition to the above scientific articles, there are many web pages on stratospheric chemistry. For example, https://personal.ems.psu.edu/~brune/m532/meteo532_ch7_stratospheric_chemistry.htm