The large hose that is located behind the mass airflow sensor is called an air intake hose.
It brings air to the engine, where it is mixed with fuel. Most cars have internal combustion engines, which require air to operate. Here is a quick breakdown of how these engines work so you can see why the air intake hose is so important:
- As the crankshaft on a vehicle turns, it pulls the piston downward, drawing a mixture of air (which has been supplied through the air intake hose) and fuel through the inlet valve into the combustion chamber.
- The inlet valve closes and the piston rises back up again, compressing the air/fuel mixture, which in turn makes it more flammable.
- A spark is then released from the spark plug, igniting the air/fuel mixture and creating a mini-explosion. The energy from this explosion pushes the piston back down and turns the crankshaft.
- The leftover gas after the explosion is then released from the chamber through the exhaust outlet. 
Without the air intake hose, your engine couldn't get air, which would throw a kink in the whole process. Leaks or cracks in the hose can also cause problems since they can limit the amount of air getting into the engine. 
If you have cracks in your air intake hose, you may be able to repair them. In some cases, however, you will need to replace the hose all together. This video does a good job of showing how to replace an air intake hose in a typical vehicle: