There used to be a notice on the site explaining that it was hosted by Storj.io, but it appears to be gone now.

I should have also mentioned that it's an OSS project, so you can self-host if you want: https://github.com/dutchcoders/transfer.sh/

Oh, and there is also https://keep.sh, which is free for files less than 500MB - they have a commercial offering too.