How can I know that all my files were correctly downloaded?
The script will verify the integrity of each file by doing what is called a "checksum" on the file, and comparing that to a checksum generated by the server. This ensures that the file on your hard drive and the file on the server are actually the same.
If you run the U1 downloader program again after it finishes successfully, it will re-check the files against the files on the server. If any files don't match, those files will be re-downloaded. And if it finishes quickly without downloading any files, you can assume that everything is intact.
Another method is to compare the file listings on the server to the file listings on your computer. You can check what is on the server here: https://one.ubuntu.com/files/
If you have another copy of your files already, you can verify the checksums yourself, though the details of how to do that vary for each operating system. Three popular methods for this are called "md5sum", "sha1sum", and "sfv". If you wish to do this, you will need to find a suitable program for your operating system. But you could also simply open the files and make sure they are what you expected.