The extensions display an estimate based on how many of the extension's users disliked the video. Youtube doesn't expose an API for getting the exact dislike count, except to the creator of the video.
I don't think they are exposed through any public API, the extensions generally just track it themselves (which of course means only dislikes from those that have the extension installed are counted: apparently they extrapolate their returned count, which does likely mean the result is biased)