Why are expectations of future inflation important to the economy?
Because it can have influence on the behavior and, therefore, on the decisions of economic agents.
When economic agents expect a scenario and, more important, when expectations appear to converge, they become way likely to change their decisions of production / consumption / saving, etc. based on that.
If prices are expected to grow rapidly, one might think that it is wise to go to the supermarket and buy as much as you can, anticipating consumption - and probably crushing its marginal propensity to save -, for example.
On the other hand, firms might delay or postpone their production, or even try to do so with salaries negotiations, so they do not get readjusted as much - which will be the demand of labor unions, as these organizations will se expected inflation as a threat that must be tackled beforehand, before it comes to be.
So we can see that expecting higher prices in the future - or, more specifically, the rate at which consumers and producers believe inflation will reach in the future is clearly important (even though I have given exxagerate examples in order to illustrate it) to evaluate the course an economy is likely to take soon.