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
7.1 (of a task or situation) need or call for (a particular person or
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.