...he didn't.
Ash decided against catching it when he learned the reason why it was snatching food and berries: because it was only trying to feed the Stoutland it lives with.
It's basically a similar situation where Ash originally opposed catching Rowlet because he thought the Pokemon would be happier with its family than with a trainer.
But even though Ash didn't catch the Litten, it was still a pretty good episode. Learning Litten's backstory was pretty sweet, and you couldn't help but feel bad for the poor thing when that Alola Persian arrived every now and then to steal the food from it. Thank goodness Ash was there to scare that Persian off, but it was even more satisfying when Litten landed the Ember on Persian and scared it off when it got burned.
In the past seven episodes, I noticed there is a theme of fatherhood going on. First you have Ash's Rowlet who was living with a family of Pikipecks, a Trumbeak, and a Toucannon, and the Toucannon was its father-figure. You can tell Toucannon cared for it like a father by sending it off to Ash when it sensed how attached Rowlet had become to him.
Then you have Litten's Stoutland. The dream Litten has in this episode shows how it sees the Pokemon as a comfort-figure similar to a father, and the Stoutland even acts as a mentor-figure by teaching Litten how to improve the strength of its Ember attack. The Ember originally starts out weak at the beginning, but as the episode progresses, its attack becomes stronger and stronger to the point it culminates in the final battle against Alola Persian.
And perhaps you could even say Kukui is a father-figure for Ash at the moment since he looks over Ash much like a dad.
Speaking of Kukui, we had a funny moment during the episode where we learn Kukui tends to sleep on his computer table.
Back to Litten, if Ash were to catch it, I have a feeling it will take some time. It would probably be similar to Ash's Sandile back in Unova, where Ash encountered it very early on, but he didn't catch it right away. After a few encounters, he finally caught it.
Next week's episode should be pretty fun now that Lillie will finally have her own episode.