This related question asks about Mission Control's responsibilities during a hurricane. But there's a more general question underneath: what do the people in Mission Control do, what are their responsibilities?
The ISS is controlled from Flight Control Room 1 of the ISS Mission Control Center, located at NASA Johnson Space Center, Houston, TX.
Here's a layout of the room with the positions of each person/team:
The yellow positions are manned 24/7:
Flight Director: Flight has overall operational responsibility for missions and payload operations and for all decisions regarding safe, expedient flight. This person monitors the other flight controllers
CapCom – Capsule Communicator (legacy name from the 60s). This is the person that talks directly to the crew. It is often another astronaut.
ADCO – Attitude Determination and Control Officer. Works in partnership with Russian controllers to determine and manage the station’s orientation, controlled by the onboard Motion Control Systems. This position also plans and calculates future orientations and maneuvers for the station and is responsible for docking the ISS with other vehicles.
SPARTAN – Station Power, Articulation, Thermal and ANalysis
- ETHOS – Environmental and Thermal Operating Systems
- CRONUS – Communications RF Onboard Networks Utilization Specialist
The remaining consoles are manned as needed. For some that is every day during crew wake hours. For others it is only when a particular operation requires that individual.
- OSO – Operations Support Officer
- ISO – Inventory Stowage Officer
- OPS PLAN – Operations Planner
- PLUTO - Plug–in Plan Logistics and Utilization Officer
- TOPO – Trajectory Operations Officer
- RIO – Remote Interface Officer (Interface with MCC-M and ATC-CC)
- SURGEON – Flight surgeon
- BME – Biomedical Engineer
- GC – Ground Controller (responsible for MCC facility equipment)
- PAO – Public Affairs Officer
- MOD - Mission Operations Directorate (Management position)
- ROBO – Robotics Officer
- VVO – Visiting Vehicle Officer
- EVA – Extra Vehicular Activity Officer
- they monitor the station's health
- they monitor the crew's health
- they support the crew in carrying out their scheduled tasks (in addition to Mission Control, a lot of work goes into building the daily schedule for the astronauts, planning experiments and maintenance, etc.).
Specialists are available to assist the astronauts for EVA, station maneuvering, robotics operations (manipulator arms, etc.), vehicle docking, logistics, etc.
Basically, the ISS needs far more than the 6 people inside it to keep it running smoothly. The MCC offloads lots of functions to the ground, allowing the astronauts to focus on the core tasks (science) and non-offloadable tasks (maintenance).