I haven't found an explanation for this expression in my dictionary, but from a few bits of info I found online I think it has something to do with "bragging"? Does it have to do with the way someone speaks only or can it describe actions too? Like "Acting extravagantly".
Here's a sentence I've encountered with said expression
Context: Big-shot guy wants a reporter to document his latest achievement of ascending the ranks of government in various forms of propaganda. He's now one of the most powerful men in the universe the story is set in. The reporter and his workmate don't like Mr Big-shot because he's a pretty scummy and corrupt guy. It's pretty outrageous of the guy to request the reporter to do this specially for him. The reporter's workmate asks if he'll undertake such a task, the reporter agrees to do so and gives a short speech about how he takes his job seriously, the workmate says " 思ったより大きく出たねぇ……。" it depends on the meaning of the sentence but I believe the workmate is talking about Mr. Big-shot.
If so perhaps the speaker is saying "bragged more than I thought" or "acted more extravagantly than I thought".