What is responsible for reviving of fallen brothers?

Luck mostly

When a brother goes down in combat, there is 2 possibilities : he can either be dead for good, or have a chance to be saved at the end of the fight.

Your brother is dead for good if:

• He dies of a fatality kill (decapitated or crushed skull)
• He is revived as a Wiederganger
• He is eaten by a Nachzehrer (while "dead" ; if he is swallowed while still fighting by a stage 3 Nachzehrer he can still be saved, provided you free him quick enough), or by the Kraken
• He is poisonned to death by a Webknecht

And I think that's about it. Now, if your brother demise doesn't fall into one of these categories, he has a chance to survive with "only" a permanent injury. I don't recall the exact probability of survival, but it should be something between 30% and 50% and it doesn't depend on your stats. The only exception is if your brother has the "Survivor" perk, in wich case the chance of survival is of 90% ! However, if that brother already has all of the possible permanent injuries, he will die (and at this point, that's mercy !)