Essentially, rather than travelling the Internet to reach an AWS endpoint, traffic is routed to the nearest Edge Location, which then routes traffic over the Amazon network to the required region.
You won't get any benefit from Amazon S3 Transfer Acceleration if your closest Edge Location is in the same AWS Region as an AWS Region. This is due to the fact that all of the traffic will take the same route.
To see if it delivers additional speed, you can use the Amazon S3 Transfer Acceleration Speed Comparison tool.