On the YouTube home page:

1.) Click your User Account Icon in the top-right corner.

2.) Select YouTube Studio

From the Creator Studio Home page, navigate to the "Subtitles" Tab

In the Video tab, select a video you would like to edit the captions for.

Click "Duplicate and Edit" in the Video Subtitles screen.

General Overview:

1.) Timing/Transcript view

2.) Edit the caption text.

3.) Adjust timing by stretching/shortening caption box and moving the caption on the audio timeline.

4.) Publish edits.

Once you have pressed "Publish edits", the subtitles will be queued for publishing, and will be available when publishing is complete.