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.