The phrasal verb "go into" has several meanings. Here it means "start". So the sentence means, "It's time for me to start my dance" (start dancing)

"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."

It depends on context. If it's politicians talking about diplomacy, it may mean something entirely different. So it is important to give the context in which you found this, or the context in which you intend to use this.