My neighbor complained about my singing.

The humor in this joke hinges entirely on a classic pun, a form of wordplay where a word is used to suggest two or more meanings, or the meaning of another word similar in sound. Here, the word "verse" cleverly plays on its homophone, "worse." The setup implies the neighbor finds the singing bad, and the punchline suggests it could be even more terrible, while simultaneously referring to the structural components of a song.

Beyond just being a sound-alike, "verse" is a fundamental part of songwriting and poetry, referring to a group of lines that form a unit in a song or poem. So, the singer is not only implying their performance could be "worse" in quality, but also perhaps more extensive, with more "verses" to endure. This adds a layer of self-aware, albeit perhaps misguided, confidence to the singer's retort, making the neighbor's plight seem even more dire. It is a relatable scenario for anyone who has ever had to politely (or not-so-politely) endure a neighbor's questionable musical talents, or perhaps even been that neighbor themselves.