It's quite simple. When you request a file to be downloaded with these services they will probably redirect you to a URL like this:
Code:
http://somesite.com/dl/1234
There will be a script that the user accesses which connects to the file host using their own premium account and starts downloading the file to the server. As the file is being downloaded, the script will push the file's data to the users browser (but it will never push more data than what's been downloaded, for obvious reasons). It can be achieved in PHP by using a HTTP header to trick the browser into thinking the data being sent is is a download, and then using a loop to send the files data to the user. It works kind of like a proxy, if it helps to think of it like that. Using this method, the user doesn't have to wait for the entire file to be downloaded into the server, as they're being sent the file as the server is downloading it.
You can also restrict how much data is being sent to the user at once by modifying the loop, for example you could send 1024kb of data to the user, and then let the loop sleep for 1 second, resulting in a download speed of 1mb/s for the user.