No. Seedr uses our 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 start a transfer into Seedr and close the site. Seedr never does anything on your device.
For example, once you've added a link, you don't need to keep your device connected to the internet, or even powered on. You may come back later and have your files immediately available on any other device ( phone, computer or tablet).
Your bandwidth is only used when you stream files on the site or download files to your device.