The short version is that Jackson took some liberties with this part of the story.
In the book, Entmoot decides to attack Isengard, and they decide surprisingly quickly:
'Hoom, hom! Here we come with a boom, her we come at last!' called Treebeard when he caught sight of Bregalad and the hobbits. 'Come, join the Moot! We are off. We are off to Isengard!'
'The Ents made up their minds rather quickly, after all, didn't they?' Pippin ventured to say after some time, when for a moment the singing paused, and only the beating of hands and feet was heard.
'Quickly?' said Treebeard. 'Hoom! Yes, indeed. Quicker than I expected. Indeed I have not seen them roused like this for many an age. We Ents do not like being roused; and we never are roused unless it is clear to us that our trees and our lives are in danger. That has not happened in this Forest since the wars of Sauron and the Men of the Sea. It is the orc-work, the wanton hewing - rárum - without even the bad excuse of feeding the fires, that has so angered us; and the treachery of a neighbour, who should have helped us. Wizards ought to know better: they do know better. There is no curse in Elvish, Entish, or the tongues of Men bad enough for such treachery. Down with Saruman!'
The Two Towers book 3, chapter 4 "Treebeard"
The idea that the Ents initially decide not to get involved is purely an invention of Jackson, and he likely did it so he could have his cake and eat it too: in the movie, we have a very poignant scene when Treebeard walks out of Fangorn and sees the devestation surrounding Isengard:
Treebeard: Many of these trees were my friends, creatures I had known from nut and acorn.
Pippin: I'm sorry, Treebeard.
Treebeard: They had voices of their own...[Looks towards Isengard; angrily] Saruman! A wizard should know better! [He roars] There is no curse in Elvish, Entish, or the tongues of Men for this treachery!
A version of this scene exists in the book (and note the similarity between the book and movie at "A wizard should know better"), but the lines are given to Bregalad, also called Quickbeam1, an Ent whose sole purpose in the story is to keep Merry and Pippin entertained while the Moot is going on.
Rather than give up this scene along with the frankly unnecessary character of Bregalad, which does a lot to get the audience sympathizing with the Ents and their cause, Jackson decided to shake up the chronology a bit and give the lines to Treebeard.
As to how Treebeard has the authority to do this, the main reason they decided not to attack was that they didn't believe the war was affecting them; as Treebeard says earlier:
Pippin: And whose side are you on?
Treebeard: "Side"? I am on nobody's "side", because nobody is on my side, little Orc. Nobody cares for the trees anymore.
And then when justifying the Moot's decision:
Treebeard: It's not our war.
Although Treebeard (and, by extension, the other Ents) are under no illusions that Saruman is good for the world, they don't yet realize that he's so bad for the forest. Seeing the wholesale destruction educates them, which of course is what Pippin was counting on.
Ultimately I have to agree with Richard's comment: Treebeard merely showed the other Ents the truth about what was going on, and that was enough to rouse them. Jackson is just taking a shortcut by not showing us the part where all of the other Ents mourn the dead trees and then shout angrily into the sky, before getting to the interesting bits.
1 So named because he once replied "Yes" before another Ent had finished asking the question. No significance to that, I just think it's a funny story and one of the many humanizing anecdotes that represent why I love these books.