The word is being used very broadly and vaguely there; the meaning is "to function".
The "rapt" audience forgets themselves and awaits your words. You decide when to speak, in your own good time, which creates "an air of anticipation", that is, puts the audience on the edges of their seats, as the saying goes.
The person on stage "sets the tempo".
P.S. The phrase on my time means "according to what suits my schedule".
I can help you move into a new apartment this weekend, but it will have to be on my time. I have some prior obligations.
That would be spoken "on MY time", the word my receiving emphasis. So on your time in this particular context would be "according to your desired pace or timing".