This is unlocked by having the cat named Sharikov enter your shelter.
You may get this randomly because it comes up relatively often and there are multiple pop-ups that bring it in.
These pop-ups and answers are:
Saying yes to a "sleazy" man that shows up with a sack and offers to trade it to you for a Soup Can
Saying yes when you find a cat in a destroyed settlement surrounded by bones (I think)
Basically just say yes to anything involving a cat
Getting Sharikov is involved in unlocking:
The Cat Lady achievement in 60 Seconds! worth 83 pointsLearn to serve and obey the new masters.