According to the website ProFlowers, the theory that plants can benefit from speech was first published in a book called Nanna (Soul-life of Plants) by German professor Gustav Fechner in 1848.
Fechner was born in 1801, and was also a philosopher, physicist and experimental psychologist. He is the founder of psychophysics, and he inspired many 20th century scientists and philosophers.
A little deeper digging revealed that Fechner held an animistic view of the world, which means he saw animals, plants and other inanimate objects as possessing some kind of spiritual essence. In light of this, his view on plants isn’t terribly surprising.
But you don’t have to search far and wide to find other books that support this view of plants responding to sound.