The Chamber's dictionary was taking you in the right direction.
The phrase, "speaks to one’s heart," has a long history in the Western Christian experience and high currency in today's religious discourse. The idea of "speaking to the heart" appears in the Bible repeatedly. See for example God speaking in Ezekiel 3:10
"Son of man, take into your heart all My words which I will speak to
you and listen closely."
Presently, there is an extremely popular song/hymn that moves across Protestant churches and takes the phrase as its title. Just a sample of the lyrics:
"Speak to my heart, Holy Spirit Message of love, love to encourage me"
Now compare that with a 18th century Bible commentary/concordance when defining the phrase:
"speak to one's heart, to comfort him, to say pleasing and affecting
things to him."
That same definition runs verbatim through various other concordances and in the mid 19th century Noah Webster picked it up and (undoubtedly encouraged by Transcendentalism) put it in the current words of the hymn:
"To speak to one's heart, in Scripture, to speak kindly to ; to
comfort; to encourage."
By now you my have noticed that the phrase "Speaks to one's heart" is not synonym with "to touch one’s heart," though they are closely related.
An exegesis of the biblical texts on this subject reveals the principal issue is the passing of information, but not any kind of information nor in any kind of language.
The verb "speaks" plays a crucial role. While "touching" is undefined, subjective and open to various interpretation, "speaking" is about connecting to the recipient with words that make sense either intuitively or by logic. The point is that whatever "speaks to one's heart" knows the language to convey the meaning and also knows (and this is as important) the information that the recipient needs to "hear."
Two important things:
1- The language (which includes the rapport and manners: treats you with respect and speaks your tongue).
2- The message (exactly what you need to hear [film "The Matrix]).
The result of "speaking to one's heart," however, is the same as "touching one's heart." The person is reestablished, energized, and reoriented (with all of its geographical implications).
In short, when something "speaks to your heart," it fills the frontal lobe of your brain with essential information and then reaches to your amygdala with tenderness.