How did Pippin know that Saruman was cutting the forest

How did Pippin know that Saruman the White was cutting the forest on that scale?

FOR THE MOVIE

He didn't

Treebeard does know that there a a great War coming. After he lets the hobbits sleep he leaves saying.

"I have business in the forest. There are many to call. Many that must come. The Shadow lies on Fangorn. The withering of all woods is drawing near."

^{THE LORD OF THE RINGS - The Two Towers - Screenplay by Fran Walsh & Philippa Boyens & Stephen Sinclair & Peter Jackson}

In fact, Treebeard already knew Saruman was doing something but doesn't seem to care.

On the way to the Entmoot to decide if the Ents will take part in the Great War... they see Isengard in the distance..

AERIAL ON: Amidst the lush foliage of FANGORN, TREEBEARD continues his own arch with MERRY and PIPPIN in tow. In the distance, ORTHANC is clearly visible in the RING OF ISENGARD...a dark shape moves across the land towards ROHAN.

PIPPIN: Look. There's smoke to the south.

TREEBEARD: There is always smoke rising......from lsengard these days.

MERRY: Isengard?

ANGLE ON: MERRY and PIPPIN reach the upper branches of TREEBEARD, just above the tops of the trees.

TREEBEARD: There was a time......when Saruman would walk in my woods. But now he has a mind of metal......and wheels. He no longer cares for growing things.

HIGH WIDE ON: On the FIELDS of ISENGARD, the FLOOD of the URUKHAI army marches.

ANGLE ON: MERRY and PIPPIN ride atop TREEBEARD, worried.

PIPPIN: What is it?

MERRY: It's Saruman's army. The war has started.

Eventually, the Ents decide NOT to go to war but rather to let it pass over them as they have always done.

TREEBEARD: The Ents cannot hold back this storm. We must weather such things as we have always done.

Treebeard tries to take them to safety by going to the Western edge of Fangorn...but that's not where Merry and Pippin want to go

MERRY: Our friends are out there. They need our help. They cannot fight this war on their own.

So, Pippin takes advantage of Treebeard's 'unhasty' nature by 'tricking' him into taking them South (since that's the direction the hobbits want to go).

PIPPIN

Yes. Exactly. If we go south we can slip past Saruman unnoticed. The closer we are to danger, the farther we are from harm. It’s the last thing he’ll expect.

But he might have suspected.

It's possible that Pippin suspected that something might be going on after this ...

MERRY (distantly) The fires of Isengard will spread, And the woods of Tuckborough and Buckland will burn. And...

MERRY turns to PIPPIN.

MERRY (CONT’D) (angrily)

...and all that was once green and good in this world will be gone.

MERRY looks into PIPPIN’S eyes intently.

Movie response only

Initially, Treebeard brought Merry and Pippin before the white wizard, who in a later scene with Aragorn, Legolas and Gimli is revealed to be Gandalf, returned from his battle with the Balrog.

From Wikipedia:

In The Two Towers, Merry and Pippin run into the Fangorn Forest in an attempt to escape a band of Uruk-hai and orcs that kidnapped them. There, they are pursued by one orc. Treebeard awakens, crushes the orc, and takes Merry and Pippin to the "White Wizard" (later revealed to be a revived Gandalf) to see if they were "little orcs" or not. Assured that they are not orcs, Treebeard keeps the hobbits with him for safety.

In The Fellowship of the Ring, Gandalf spends a considerable amount of time captured by Saruman atop the tower of Isengard (where he later escapes via giant eagle), so he has a perfect view of what Saruman has been doing.

Gandalf tends to speak wisdom to people when they part ways. To Aragorn, he said to look at the east/west (I forget) at the fifth day, because he knew he'd be back with reinforcements by then.
If Gandalf parts ways with Merry and Pippin and leaves them with the Ents who would be powerful allies but are currently apathetic to the war at hand; it stands to reason that he would tell Merry and Pippin of Saruman's deforestation; so that they can sway Treebeard's mind.

It's also possible that Gandalf simply mentioned it when talking about Saruman; and Pippin connected the dots himself.

So here's what we know:

• Gandalf is aware of the deforestation because he was Saruman's prisoner from the point where he revealed his evil intentions. He was either already clandestinely deforesting, or he started when Gandalf revealed his intentions. Either way, Gandalf's cell on top of the spire was a perfect viewing location to see the deforestation and everything else.
• Merry and Pippin meet with Gandalf almost directly after meeting Treebeard, since Gandalf needs to reassure TB that they aren't orc spies.
• Gandalf, for some unexplained reason, decides to leave Merry and Pippin with Treebeard. For an equally unexplained reason, the Ents are now having an Entmoot and discussing going to war. It is very likely that Gandalf was aware of this when he was with TB, Merry and Pippin; if not the sole reason why the entmoot is taking place at all.
• Pippin seems to think of sending TB close to Isengard on the spot, he has a revelation. So the most likely explanation is that he knew about the deforestation (presumably through Gandalf), but hadn't yet thought of using this as a way to sway Treebeard's intentions for war (neither did Gandalf, apparently).