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

> 思ったより大きく出たねぇ……。

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".
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2017-07-19T20:47:55.067
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meaning expressions idioms
Meaning of the expression "大きく出る"
The idiom 大きく出る is not bragging, but talking about something exaggeratedly as
if it were more important than what it actually is. In other words, it's
refers to making a mountain out of a molehill.

In this context, the workmate apparently expected a casual and simple answer
(something like "Sure, why not?") but actually ended up listening to a serious
speech about the reporter's philosophy. 大きく出る refers to this fact.
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2017-07-20T01:41:54.743
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> “大きく出る”
>
> "bragging"? Does it have to do with the way someone speaks only or can it
> describe actions too? Like "Acting extravagantly".

I would put it as _to take a bold step_. It can be used for either a word or
an action. Bragging can be included, but it doesn't have any negative feel,
and can be also used for one's own action such as ここは少し大きく出てみてもいいかもしれないですね。 In
your context, I think it's a remark expressing the unexpectedness in a good-
hearted manner.
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answer