Updating an old movie to HD?

Film is re-scanned digitally at a higher resolution. The quality is greatly improved, and on occasion they’ll actually do a frame-by-frame digital restoration to adjust colors that go out of balance, remove dust and scratches, etc.

They often have the uncompressed original files somewhere, or they will take the physical film copy, bring it onto a computer, clean it up or make the colors vivid, etc, and output that onto the DVD.