Stage
noun
A dance machine isn't really a machine... a 台 is a large machine thing counter, so in this case it's talking about a stage. A dancing stage is a stage.
ぶたい
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own. Just remember that 台 is read as たい here, not だい.
すごくいい声持ってるね。舞台で歌を歌ってみようと思った事はないの？
You have such a great voice. Have you ever thought about singing on stage?