First of all, Windows 10 CANNOT and DID NOT download itself.
Windows 10 can and does get automatically downloaded and installed, unless additional actions are taken to prevent it from occurring.
When KB3035583 is installed, (the one that adds the Get Windows 10 nag prompt) WIndows 7 and Windows 8/8.1 Will download Windows 10 install files automatically in the background through BITS. (the background Intelligent Transfer Service). BITS checks for some files KB3035583 creates, and if present will attempt to download the Windows 10 installation files. This takes place regardless of what answers/interaction took place with the Get Windows 10 App.
Further, An update to Windows Update itself was made such that Windows Update will look for those install files downloaded by BITS. For some time it would add "Upgrade to Windows 10 Home" or "Upgrade to Windows 10 Pro" as an optional update if the install files were found. This has since been reclassified as a "recommended" update. This means that any user that uses the recommended option of Automatic updates will have Windows 10 automatically download as well as have it's installation started. Microsoft has also released a steady stream of patches to the "Windows Update" tool itself, these patches are designed to unhide KB3035583 if the user has accidentally gone in and made it hidden, since obviously nobody would ever do that intentionally.
The installation process is itself hands-free. What happens is a window appears saying "Your Windows 10 Upgrade is ready" and in addition to the usual babble about how it makes you do your work faster, it states that Windows 10 will be installed in:" and a timeout. This can be cancelled (Postponed) at which point it will appear later. If there is nobody at the computer when this occurs, the timeout will expire, and Windows 10 *will* be installed automatically.
In fact, the very problem the OP is describing has been reported quite a number of times. Windows systems where users have gone with the Microsoft-recommended option of automatic updates will often find that not only has their system been "upgraded" to Windows 10, but it doesn't even boot properly anymore. There are many cases of boot loops or otherwise unbootable systems after this automatic upgrade process.
Unfortunately while the Windows 10 upgrade allows rolling back within 30 days, that is only a viable option if the system is not booting, which is not the case in some cases. The worst part is there is no proper fix for this problem to get from the broken Windows installation to either a working Windows 10 (or the previous version) install; it requires either offline adjustments to the registry via another PC editing the hive files directly, or a full Windows reinstall to resolve the issue. Microsoft has not yet provided any proper resolution to customers affected by KB3081424, which is one of the components that gets installed during an upgrade which seems to have the potential to make the OS unbootable.