"It takes" is an idiom that refers to the skills, character, or other qualities necessary to achieve something.
"It takes a bigger man to walk away from a fight" means that the man who can walk away from a fight is superior in morals, judgment, intelligence, etc.
ell.stackexchange.com/users/111962/SarahT
Answering directly your first question: It takes what?
According to the Free Dictionary what it takes The necessary expertise or qualities
She's got
what it takes to make a good doctor Inherited wealth is what it takes to maintain that lifestyle.
This idiom uses what in the sense of "that which" and take in the sense of "require."
The linked source extracts the text above from The American Heritage Dictionary of Idioms by Christine Ammer
According to the Oxford Dictionary take 7.1 (of a task or situation) need or call for (a particular person or thing)
It will take an electronics expert to dismantle it Notice the distribution of the elements in the example, that match the ones in your question:
It takes - It's required
What?
An electronics expert / A bigger man
For what?
To dismantle it /
To walk away from a fight
The comparison being made in your sentence, the one that uses " than ", is between the man who avoids the fight and the one that stays.
The first is bigger morally speaking.
Let me rephrase your sentence: A more qualified man (First Man), morally speaking, is required for avoiding a fight than the one (Second Man) that stays.
First Man is bigger than Second Man.
And last but not least, notice that we are in front of a kind of proverb : a short, well-known pithy saying, stating a general truth or piece of advice .
ell.stackexchange.com/users/48962/RubioRic
What the sentence is really saying is, "A man who refuses to fight is braver than a man who stays and fights."
The sentence structure is the same as "It takes a hotter furnace to melt iron than it does to melt tin."
In other words, the furnace which is needed in order to melt iron is hotter than the furnace which is needed in order to melt tin.
I admit that the exact phrasing is a little strange.
Why would we write something like "It takes a braver man" instead of "It takes more bravery"?
Maybe there isn't a good reason.
But that's the way the saying goes.
ell.stackexchange.com/users/2020/Tanner Swett
The meaning of take in the sentence is to require or to need : to need : Parachuting takes a lot of nerve.
Transitive verbs take a direct object.
[ + -ing verb ]
His story took some believing
(= was difficult to believe).
Big man is an idiom; a big man is exceptionally masculine (usually implied to be tough and strong).
The sentence is saying that, to start a fight, you must be a big, tough, manly man.
The than in the sentence compares the "big man" who would start a fight to a "bigger man" who has the wisdom to avoid the confrontation.
ell.stackexchange.com/users/18196/Harrison Paine
This is actually a normal comparison with than .
The phrase should actually be simplified for understanding the structure:
It takes a bigger man to walk away from a fight than it does to stay and fight
It takes more now than it took in the past
In both cases, the structure is roughly: [subject]
[verb]
[object in comparative] than [subject] [verb].
and it is equivalent to phrases:
He plays football better than I do
He is taller than me ( = ...than I am )
The only difference is that in our sentence, the comparative does not depend directly from the verb, but rather from a noun and it is that noun that depends from the verb (it is verbal object).
ell.stackexchange.com/users/113544/Eleshar
I found this sentence reading. But I didn't get the meaning. Could you explain it to me?  The sentence is:    It takes  a bigger man to walk away from a fight  than  it does to stay and fight   It takes what? And it takes more or less? Because we have "than" so more than or less than ... I need an explanation because it seems like something is missing in this sentence. Thank you.
It takes a bigger man to walk away from a fight
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ell.stackexchange.com/questions/245096
ell.stackexchange.com/users/112020/slim Hass
"It takes a bigger man to walk away from a fight" means that the man who can walk away from a fight is superior in morals, judgment, intelligence, etc. “It takes" is part of the idiom and refers to the quality required to achieve something. 
“It takes" is part of the idiom and refers to the quality required to achieve something. 