"He gave us the signal to begin."
Translation:Él nos dio la señal para empezar.
Can someone explain why we are using para or de instead of "a" before empezar?
Because to dar la seña means "giving a description or password". The word more refers to something distinguishing, a "distinguishing characteristic" rather than a gesture or indication.
HOWEVER: were the phrase "He made the signal..." then then "Él hizo la seña para empezar." would work because then you are talking about a distinctive gesture.