My understanding of filter representations is that it's used to estimate the variance of forecast.  We can convert the φ parameters into an infinite series of θ parameters.  This is helpful because the θ parameter only changes one period in the future.  I'm anticipating that the final might ask, "What is the variance of the one period ahead forecast?  Two?  Three?" 

There's a few practice problems in the Module 7 Stationary mixed processes that might be helpful in understanding the concepts.

[NEAS: Correct; the filter representation converts an autoregressive processes into a moving average model of infinite rank. This simplifies the formulas for the variance of forecasts because all the error terms are independent, whereas the observations are serially correlated.]