Edit: This was supposed to go up at midnight, I am unsure why it didn't go through. Sorry folks!
Run character driven or heavy story games long enough and you're going to end up having a few intense sessions on your hands. Have players who like to delve into the psychology of what makes their characters tick and you're definitely going to have it happen. So what do you do when the metaphorical shit hits the fan and everyone's emotions spike? You ride it out, but you also take time after the scene and after the game to make sure everything is cool.
Emotion At The Table
Emotions at the table are perfectly fine. If those emotions stem from in character actions, thoughts, plans, and beliefs all the better. Emotion is a natural response from tension. I've had people cry at my table and it has been both awesome and terrible depending on the situation. Those sessions have lived on in memory regardless of which it was though, and the reason is because the emotion amped everything up to the next level.
Still, for some groups - and especially newer GMs - the question remains on how to handle it. Which is what this post is about.
Trust Requires Communication
Communication and trust are widely touted as the big two factors for any relationship to work. The fact is though for trust to exist you need to communicate. If you never tell someone anything, they have no reason to trust you. The same is true in your gaming group. In particular you need the following two things:
1 - your group must trust each other (including you) enough to let the walls down and have emotion be a factor at the table. If this isn't true, don't try for emotional sequences because without that trust they're either going to fizle and fail, or blow up in your face.
2 - Your group must trust that if they communicate discomfort to you that the scene in question is going to stop.
In my opinion 2 is the most important thing. If someone is uncomfortable whatever is causing that discomfort needs to stop. Now, there is a difference between someone being truly uncomfortable (as in the player) and a character getting the creeps. Most gamers know this difference, but even if they don't it doesn't change the rule. When a player is uncomfortable you stop. Otherwise it becomes unfun for them and can cause damage to your relationship.
Ride Out The Scene
If you have those two things, when emotion does rear its head you can ride the scene out. Keep track to make sure emotions are in character. Make sure no player is outbursting with their own emotions or getting too loud for your play area. However, let the scene go on. If someone is uncomfortable they can and should tell you. If everyone is playing along, it should be ok to continue. Let the scene play out, and then...
Check To Ensure Everyone Is Ok
After an emotional scene make sure your players are having fun. The gap between scenes works best because it lets the players have a break from their characters while the game is in transition. This is also a good time to ask if anyone needs a break to get a drink, use the bathroom, or any other excuse that could let them be alone - or with whomever they are closest with at the table - and get their head back in order. Even if the emotion was purely in character, it can take a moment to amp down and release the energy. Give your players that.
Talk About It After Session
Finally, and this is why the post is titled "aftercare" talk about the scene after the session. Talk about it as a group, and talk about it with the individual players. Make sure people had fun (OOC) and enjoyed the scene. Make sure people are ok with that sort of thing in the game. Let people talk to you if they have concerns. It is possible that someone was uncomfortable during the scene but didn't want to kill the fun for the group. Give that person a chance to tell you after the game so you can address it.
Above all, if things played out bad in the scene give folks a chance to talk out their character's side, and where they were coming from. Perspective can do a lot to help understand what happened - especially for the person who came out the worst in the matter. It also gives people a chance to see that nothing that happened was OOC or about their relationship with the other player outside of the game.
These conversations can be done after any session - and they can and do add a lot of depth to play as people share opinions - but after an emotionally heavy session they can be particularly important. Let folks vent. Let folks re-center themselves, and then call it a night.
Remember, it's a game and supposed to be fun. When it gets heavy and tense, make sure you don't lose the fun.