In order to get this achievement you have to trade rations to the hungry family when they appear 4 separate times. The event occurs near the beginning of the day when you're told if you have new quests. Simply accept the offer when it's given. Sadly this is quite an RNG achievement. In my first playthrough they only showed up once. In my second playthrough that I'm doing now, they've shown up 2 times and I'm at day 30. I likely won't get the last 2 in the next 12 days.
Something I haven't tried yet but intend to is making a cloud save right before finishing a day, and then finishing the day and seeing if the event happens. If it doesn't, I would delete my save and then resync with the cloud and try again to see if I get different events.
You can abuse cloud saves to get the event but I could never seem to get the event multiple days in a row, so if you're super deep into a playthrough already and you aren't close, you're probably out of luck. This is something you want to do early on just to be safe.