No. Seedr uses it's own cloud to add files directly from the internet to your account. That's how your files get there so fast (Up to 100MB/s). That is also why you can tell us to download and even close the site - it never does anything on your computer!
For example, once you've added a link, you don't even need to have your computer on. You may even come back later and have your files immediately available on any other device (whether phone, computer or tablet).
Your bandwidth is only used when you stream files on the site(Like on YouTube) and if you choose to download your files to your computer.