This is a step-by-step explanation of Brining Down the Pain that I came up with for my group, after lurking on this forum for a while.  Maybe it'll be helpful to somebody.

Bringing Down the Pain

If you lose a contested roll, you can Bring Down the Pain.  If you do, you use the system described here to resolve the conflict instead of the usual one-roll method.  Only the players of protagonists can Bring Down the Pain.  This system is similar to other game's combat systems, but it can be applied to any conflict.  Instead of rolling once to decide the whole conflict as usual, you go into a blow-by-blow description of events.  Use it to "zoom in" on the conflicts that are important to you.  It works like this:

Step 1:  Someone loses a contested ability roll.  You'll want to remember how much they lost by - it'll matter in the first round of BDTP.

Step 2:  Someone involved in that roll Brings Down the Pain (the Story Guide can't).

Step 3:  Declare intentions.  An intention should be clear and allow for multiple types of actions to be used in accomplishing it.

Step 4:  Decide on your character's action and what type of action it is.  Mechanically, actions fall into four groups.  Perpendicular actions are in direct immediate conflict with your opponent.  Each character's action inhibits the other character's ability to perform their action.  Parallel actions are not in direct conflict.  Their goals may be in conflict, but neither action inhibits the other action's success.  Defensive actions try to stop other actions from harming your character, and allow your character to regroup and change intention.  Supplementary actions don't help you this round, but help the action you intend to do next round.  Like this:

There may be a dispute in which two or more players keep changing their action in response to each other.  If this can't be resolved through negotiation, the player who wants to take a perpendicular action must take a defensive action.

This step is commonly called the "free-and-clear" stage (a term from Ron Edward's Trollbabe), because events that are hidden to certain characters should still be revealed to those characters' players.

Step 5:  Everyone rolls the relevant ability, which is Endure, React, or Resist if their action was a defensive action.  If this is the first round, the winner of the ability roll that started BDTP gains bonus dice equal to that roll.  Use the results of the rolls to narrate the results of the characters' actions.

Step 6:  Determine harm, if any.  This is kind of a part of step four.  Harm is not necessarily physical.  It could be pretty much anything that makes it harder for you to get what you want; from injuries to a crushed reputation.  Harm is usually associated with the pool that the ability that caused it is associated with, but if everyone agrees, then it could be associated with another one.  Harm does not "fill up"; you could have the box at two checked, and not have the box at one checked.  Harm at a level that's already checked gets bumped up to the next level that's not checked.

Perpendicular Action
Roll the appropriate ability.
Subtract the loser's successes from the winner's.
The loser takes harm at that level.
Parallel Action
Roll the appropriate ability.
Do NOT subtract successes.
Both characters take harmat the level of their opponent's roll.
Defensive Action
Roll Endure, React, or Resist.
Subtract the loser's successes from the winner's.
If your OPPONENT won, you take harm at that level.
If YOU won, add that many bonus dice to your next action.
You may change your overall intention for the conflict.
Supplementary Action
Roll the appropriate ability.
Do NOT subtract successes.
If your OPPONENT won, you take harm at the level of their roll.
If YOU won, add that many bonus dice to your next action.

If you took harm at one, two, or three, you get one penalty die to your next action in BDTP.  If you took harm at four or five, you get one penalty die to all actions associated with the pool that the harm is associated with.  If you took harm at six, then you must spend a point from the associated pool to use any ability of that pool, and that action still has a penalty die.  If you take harm at seven or higher, you must give, and let the other side have their intention.

Step 7:  Return to step four.  At this point, you have the opportunity of voluntarily giving, and letting the other side have their intention.  It's often a good idea to give before your opponent changes to a more severe (or deadly) intention.

Steps 4-7 repeat until someone wins, and then you return to the normal game system.

Some general notes:

Ties.  In the event of a tie on a perpendicular action, no harm is done, and both players have the opportunity to change their intention.

Weapons.  Because BDTP applies to more than just violent conflict, more than just violent items can be weapons.  A weapon's bonus ranges from +1 to +3, and gets more specific the more powerful it is.  A weapon can have multiple bonuses.  For example, a whip may give +1 in combat, +2 when used on members of the Zaru culture, and +3 against one specific Zaru revolutionary leader.  The bonus applies to harm, not to the roll, so weapons make you more dangerous, but they don't directly help your chances of winning.  Almost anything can be declared a +1 weapon.  To get anything bigger than that, you have to take the Secret of Imbuement, which lets you make the item more powerful in other ways as well.

Healing.  A character can roll First Aid or Counsel to heal another character.  Erase the check at the level of the success on the roll, or the highest one below that.  So, if the medic rolled a five, they would heal the highest level of harm up to five, but couldn't heal harm at level six.  As an optional rule, characters may heal themselves by spending associated pool points equal to the level of harm.

Shaking Out.  As an optional rule, you may decide that all harm "shakes out" after BDTP.  Under this rule, harm at one, three, five, and six would become harm at one, two, three, and four.

Multiple Opponents.  There are two methods of using BDTP in a conflict with more than two characters.

The first method is for when everyone has their own intention.  When a player declares their intention, they must also declare which character they are affecting.  They may only damage that character.  You must change intention to change what character you are affecting.  BDTP ends when only one character is left standing.

The second method is for when a group of characters all have the same intention.  Use the group ability check rules to distill the group's ability rolls into one roll (it's essentially the same as a bunch of supplementary actions).  Any character can spend pool points to help any other character in the group.  The winning side distributes harm among the group as they see fit.  If a group member takes harm at six or higher, they drop out of BDTP.

Thanks for the clarification, Dan. I think I understand BDTP better now.

Regarding Healing of yourself or someone else: Can you do this during BDTP, or can it only be done after the conflict is resolved?