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My memory's a little fuzzy, but I believe this is how the information was obtained: After the speech the White House had Kerry investigated (on this see Nixon Thought John Kerry Was a "Phony"), and if I recall right that was one of the things they learned. Subsequently Kerry was confronted with the charge--I believe by John O'Neill during a debate, though I'm not sure. Walinsky denied the charge, but my recollection from what I read was that I found his defense unpersuasive. This is discussed in Brinkley's bio. Here's an FR thread which quotes the same information from Unfit for Command:

Are Kerry's records really open?

Unfit for Command [Excerpt] Kerry's testimony to the Fulbright Committee was a carefully orchestrated piece of political theater. Fulbright wanted a presentable, young Kennedy-esque face to put on the antiwar effort, and Kerry wanted a national forum from which to launch his climb to political celebrity. Ted Kennedy helped arrange Kerry's testimony with Senator Fulbright at a private fundraising event held at the home of Democratic senator Philip A. Hart of Michigan.

………Once Kerry learned that he would have the chance to give testimony before the committee, he recruited the assistance of Adam Walinsky, a speechwriter noted for his work with Robert Kennedy. Walinsky drafted the speech and coached Kerry on its delivery. The only image Kerry wanted us to see was a myth: a young man with a burning passion for the truth, the leader forced to sleep on the ground, the man answering his country's call to be where he was urgently needed, before a committee of the United States Senate where the senators and America were urgently waiting for his firsthand criticism of the war. He proceeded to level his charges: [End Excerpt]