This is the 9862nd most frequent Japanese word.
触り
"触り" means "touch" or "feeling of contact."
Here, "触り" refers to the tactile impression or texture of the fabric.
この布の触りは滑らかだ。
The feel of this fabric is smooth.
In this instance, "触り" is used to signify the main point or essence of the story.
彼の話の触りだけ聞いてしまった。
I only heard the gist of his story.
Used as a verb derivative here, "触り" relates to the action of physically touching the toy.