"MY dance" means a prepared, choreographed routine, part of an act.
"GO INTO my dance" means begin dancing.
It is proverbial in the theatre that dancing raises the emotional level. In a musical comedy, "when the emotion becomes too strong for talk, they sing; and when it becomes too strong for singing, they dance."
So, an actor performing a vaudeville routine might sing a song or so, and get some applause. "I win them over by singing, but I really get them at the end when I go into my dance."