What is a verb meaning "bringing out emotion within someone?"

You could say that it stirs emotion(s).

( This is in the sense of awakening, not mixing up.)

Something can evoke a certain emotion or response. "Evoke" is defined by dictionary.com, for instance, as "to call up or produce (memories, feelings, etc.)"

to flush out emotion

means to bring to light emotion

Quality Teaching in a Culture of Coaching - Page 62 Stephen G. Barkley - 2010

... yet a feeling remains just what the person feels. It's neither right nor wrong, appropriate or inappropriate. It's just what's so. As coaches, we can flush out emotion by gently and objectively confirming what we heard (even if we "heard" it

where:

flush out

transitive verb

: to bring to light : make public or available

Merriam-Webster Unabridged Dictionary