It appears, from the Lord of the Rings books, that a wizard's staff is a method of channelling their respective energies.
For instance, in The Fellowship of the Ring, Book 1:
..."The figure lifted his arms and a light flashed from the staff that he wielded. A mighty eagle swept down and bore him away."...
In Book 2:
..."...he thrust the end of his staff into the midst of it. At once a great spout of green and blue flame sprang out,..."
There are also many mentions of Gandalf lighting the way through Moria using his staff, and guiding the party (using the staff as a beacon).
As pointed out by Victim of Circumstance, Gandalf strikes the bridge with his power, most likely channelled through his staff to a point of force at the staff's end, so much power that it destroyed the staff. user1643723 is absolutely right in his comment - there is a power there, but maybe not within the staff itself.
Later, in The Two Towers, Book 3, we find Gandalf in Edoras. Háma states:
"The staff in the hand of a wizard may be more than just a prop for age."
There are many more references to the power of the staff throughout the books (a Kindle is a great resource!).
So, it could be that the wizard's staff is more than just an emblem of power - it could be a conduit to channel such power to a fine point, or a focus to bring that power to bear.