Amsien is reward available in the quest Stolen Property. While his role in the quest is minor, completing the quest may allow the player to have further interaction with the Khajiit, depending on whether he was allowed to escape.

Amsien the Housecat
Amsien can be met at different points of the quest depending on the player’s choices. However, if he is freed and manages to survive, he will travel to Riverside Shack where he will ask the player to take him to Darkwater Crossing. After some time, the player can ask Amsien how he is faring, at which point he/she can recruit him as a house servant.

– The time between Amsien arriving at Darkwater Crossing and becoming available as a housecat is done through his greeting, “Sugar and Sands, friend.” Upon saying it twice, the stage will set to 40 and enable the dialogue.  However, as the greeting can only be said once every 12 hours, it guarantees the player will have to wait a minimum of 24 hours before recruiting him.
– It is possible for Amsien not to be freed by visiting Rothvine Manor and using stealth. Traveling to Shattered Plank will always result in Amsien being rescued.