What is the meaning behind the quote "Poetry is plucking at the heartstrings, and making music with them.”
Poetry always speaks to each reader in a different way. We are each unique, and as such, we bring a totally different set of experiences "to the table" when we read a poem. It is the same way with quotations, and sometimes it can be difficult in that the quote may come to us without any context (other writing) to provide hints. I can give you my impression of what the line means.
The first thing I notice, though it does not lend itself to understanding the content, is that the author of the quote (Dennis Gabor) uses personification to describe poetry in terms of a person—able to pluck, something usually done with a stringed instrument. And so this line is a metaphor, comparing poetry to a musical instrument, and it's a logical comparison because poetry is, by nature, so musical.
Poetry is plucking at the heartstrings, and making music with them.
I believe this means that poetry speaks to our hearts. "Plucking at the heartstrings" refers to eliciting an emotional response from the reader—but that the response is not like a single note, but creates an abundance of response in the reader, like that of an entire musical composition—a complete and beautiful song—where reading anything else does cannot appeal to the feelings of a reader in the same manner. With a poem, the experience is often like listening to music, for poetry appeals to the reader on many levels, but literally, it is most often written (especially if there is rhyme and poetic devices like rhythm, alliteration, etc.) to sound musical, as poetry has always been meant to be read aloud rather than read silently from a page.
Consider "My Papa's Waltz," by Theodore Roetke, which sounds musical, but can also bring to mind an emotional response of nostalgia in many readers:
"My Papa's Waltz" The whiskey on your breath
Could make a small boy dizzy;
But I hung on like death:
Such waltzing was not easy.
We romped until the pans
Slid from the kitchen shelf;
My mother's countenance
Could not unfrown itself.
The hand that held my wrist
Was battered on one knuckle;
At every step you missed
My right ear scraped a buckle.
You beat time on my head
With a palm caked hard by dirt,
Then waltzed me off to bed
Still clinging to your shirt.
One source describes the poet's ability to use words to create music, referring to the creation of "verbal music."
The poet of verbal music constructs a musical pattern of words that becomes a source of delight, that moves us as a melody does.
It is the emotional response to melody that is compared to the same kind of response to words in a poem: in the "plucking at the heartstrings."
To me this means that when you first become a poet it is like you are walking around in the dark trying to find the parfect place to fit in. After a while light starts to appear and then one day you are standing in the light. When you write poetry you have to make people feel something using string emotions and words. That is what makes people want to read poetry.