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Left My Heart - Introduction
If you're neutral with Caesar's Legion, then you have a few extra choices. People that are already hostile with Caesar's Legion will only be able to go the rambo path for saving the slaves.
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Left My Heart - Saving the Weather's Family
If you want to do this peacefully (possibly because you plan to irradiate Cottonwood Cove in "Eye for an Eye"), then you'll need to talk to Canyon Runner. He's usually out guarding the slave pen or inside the HQ building next to the pen.
If you have the spare caps, you can just pay him 300 caps and get the entire family. He'll admit that the only worthwhile slave of the bunch is the young girl, so he won't push for much more. If you have more than 50 points in Speech, you can talk him down to 150 caps. With your new keys in hand, just walk out to the slave pen and talk to them to set them free (or pick a nefarious option for a laugh/roleplaying).
If you aren't on good terms with the Legion and still want to save the Weather's family, then I suggest you go ahead and do this as your way of doing "Eye for an Eye." Bring Boone with you for poetic justice and storm the camp. There aren't that many good guards inside. Most of the men are inexperienced and will be poorly armored in comparison to you at most levels. A large number also seem to want to stick to melee combat. This makes it pretty easy to just settle in and pick off the men inside the camp (especially if you have Boone with you).
Loot the keys off of Canyon Runner's body. These keys both open the slave pen and disarm the collars. If you want, you can just hand the Weathers the keys and they'll all free themselves at once.
If you can't find Canyon Runner or can't clear out the HQ building yet, then you can pick the lock on the slave pen. Once inside, you can just use an Explosives skill of 50 to disarm their slave collars.
As mentioned before, there are some nefarious options if you want to be evil. You can beat a Speech check of 50 to convince them that the collars are disarmed, when they aren't. This will make them run off and explode as they reach the camp boundary. You can also just use your Explosives skill (if it's above 35) to detonate the collars prematurely).