The sentence "My writing skills drive Li Shangyin, Wen Tingyun and others are often very busy" means that my writing skills drive Li Shangyin, Wen Tingyun and others to be very busy.
He Zhu compares himself to the writing master of Li Shangyin and Wen Tingyun, saying that he is knowledgeable and memorized, and is good at integrating the poems of his predecessors. In other words, his writing can drive the poetry creation of Li Shangyin and Wen Tingyun, just like driving his own slaves.
In short, the meaning of this sentence is to praise He Zhu's poetry creation and ability to control language.