How can I apply changes that I made in audio to video in Adobe Audition?

You need a video editing program like Adobe's Premiere to do that. Audition is for editing audio only. However, as you have found, it allows you to play a video so that you can synchronise the audio to picture. But you would normally edit the video first cutting out all the bits, including audio, that you don't want. When the video is complete you bring it into Audition to manipulate the sound, add effects and music to the original video soundtrack.