The price of a bond (and therefore the price of a bond futures contract) depends on the future path of interest rates (Which have a knock-on effect on other markets), and as central banks (CBs) set rates, speeches from central bankers may contain clues as to what that path might be.
There is evidence that markets move more on the Fed statement than the Fed decision! So, what they say is clearly important. For example: Companies used to borrow from banks. Now they borrow via the capital markets (i.e. by issuing debt). The upshot of this is that central banks have less direct influence on the economy via interest rates than they used to. This is one of the reasons for the increasing importance of central bank communication – it is another channel for them to affect the economy.
Trading opportunities from central bank speeches arise infrequently, but can offer very favourable risk/reward.