We've outlined the options to import your historical Garmin data in this article.
Other tools for transferring Garmin Connect data to Strava:
This Chrome Extention is useful for bulk selecting files on Garmin Connect for download.
The below 3rd party tool is currently out of service.
If anyone is interested, I wrote a barebone web application to download all your rides from your Garmin connect account by just providing your login. It downloads all rides from Garmin, zips them up then sends it to the browser. The process can take a while depending on how many rides it has to download. I've only tested with my account with about 300 rides and it seems fine.
I suggest changing your password temporarily for use on this web application to download your rides, then change it back after you're done if you really want to be cautious about your password.
3/1/2014 - updated to work with the Garmin authentication change