iMovie Help: Transition still keeps the previous clips' audio
Generally speaking, a transition will start before the end of a clip.  You need enough video to have something to play while the transition is occurring.  The video will normally keep running until the end of the transition and that also means that the audio will keep running until the transition finishes.  You would need to play with the volume adjustments to bring down the level of the track that you don't want to have keep playing.