YouTube offers automatic captions on videos and the ability to read those caption transcripts, but no easy way to grab the entire caption set for a video. However, you can get a transcript with a little URL manipulation.
Google Operating System notes that if you substitute the ID for a video in the following address, you'll get an XML file of its captions:
On its own, the XML can be hard to read, but the blog also helpfully points out that you can use the Online XML Reader to clean it up.
Download YouTube Captions [Google Operating System]