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Explaining the singability of lyrics is an important but missing ability of language models (LMs) in song lyrics generation. This ability allows songwriters to quickly assess if LM-generated lyrics can be sung harmoniously with melodies and helps singers align lyrics with melodies during practice. This paper presents XAI-Lyricist, leveraging musical prosody to guide LMs in generating singable lyrics and providing human-understandable singability explanations. We employ a Transformer model to generate lyrics under musical prosody constraints and provide demonstrations of the lyrics' prosody patterns as singability explanations. XAI-Lyricist is evaluated by computational metrics (perplexity, prosody-BLEU) and a human-grounded study (human ratings, average time and number of attempts for singing). Experimental results show that musical prosody can significantly improve the singability of LM-generated lyrics. A controlled study with 14 singers also confirms the usefulness of the provided explanations in helping them to interpret lyrical singability faster than reading plain text lyrics.