Ubuntu's Secure Boot support has changed over time, and the quote you've provided is outdated. Broadly speaking, there are several stages at which validation might occur:
- Validation of the boot loader -- This is the limit of what Secure Boot, per se, does. Secure Boot prevents the UEFI from launching unsigned (or improperly signed) binaries, and that's basically it. Under EFI, boot loaders are EFI binaries, so Secure Boot stops them from launching if they aren't signed. A boot loader, however, is designed to launch a kernel, which is not an EFI binary, so by itself, Secure Boot cannot block the launch of an unsigned kernel if the boot loader is designed to permit it. (There are some caveats and exceptions relating to technical details, though.) Thus, at this level, Secure Boot can prevent the boot loader from being compromised, but it can't prevent subsequent parts of the software stack from being compromised.
- Validation of the kernel -- To improve security further, the kernel should be validated. The last I heard (which was two or three years ago), the stock GRUB did not validate the kernel in any way. Thus, if you were to use a stock GRUB, it would happily launch any kernel you fed it. Ubuntu through at least version 12.04 used a GRUB that worked in this way. Modified GRUB binaries that require a signed kernel are available, and are used by many distributions. I don't recall if Ubuntu 14.04's GRUB required a signed kernel. Ubuntu 16.04's GRUB definitely requires a signed kernel, at least with its default configuration. Also, since at least version 12.04, Ubuntu has provided kernels that were signed with Secure Boot keys, and that could therefore be launched with a boot loader that did this validation. Thus, by swapping out the stock GRUB, even Ubuntu 12.04 could be configured to require a signed kernel.
- Validation of kernel modules -- Linux kernels can load kernel modules, and if a kernel is signed but loads a malware-infested kernel module, the signed state of the main kernel is meaningless. Although I'm not 100% sure of whether Ubuntu 14.04 required a signed kernel, I am reasonably confident that Ubuntu 14.04 did not require signed kernel modules. This has changed with Ubuntu 16.04. This change has been an inconvenience to people who want to use third-party proprietary drivers, such as those for VirtualBox or video drivers for ATI/AMD or Nvidia chips. These driver modules are not signed by Canonical, which necessitates adding a Machine Owner Key (MOK) ans self-signing the modules, if Secure Boot is to be kept enabled.
- Validation of user-space binaries -- Once the kernel has loaded, it launches user-space binaries -- everything from
bash
to Web browsers and word processors. In principle, a kernel could require these binaries to be signed. In practice, I don't know of any support for such a signing requirement in Linux. Such a requirement would be quite onerous, particularly for software developers or anybody who wants to run binaries from third parties.
In addition to these four major levels of (potential) verification, there are several twists, in-between levels, exceptions, etc. For instance, a kernel can include an EFI stub loader, which turns the kernel into its own boot loader, thus blurring the line between steps 1 and 2. I've also seen kernels that refuse to launch if they're launched from a boot loader that doesn't do Secure Boot validation, which turns around the requirement of step #2. In principle, validation beyond the boot loader could use a mechanism other than Secure Boot keys. In practice, in the Linux world, Secure Boot keys are used through step #3; however, it's my understanding that the kernel provides its own code to validate kernel modules (step #3), so technically the Secure Boot code in the UEFI is not involved past step #2. (I've never investigated this in depth, though, so I might be mistaken about this detail.)
Also, I've focused on LTS releases in my descriptions above. Some of the security changes may have been introduced in interim (non-LTS) releases; I don't recall the details.
As noted, Ubuntu has been increasing its level of validation requirements over time. The quote you've provided was accurate for Ubuntu 12.04, and perhaps 14.04, but not 16.04 -- at least, not with a default configuration with Secure Boot enabled in the firmware. (You could of course swap out Ubuntu's GRUB for an older version or another boot loader that does no Secure Boot validation at all.) For some applications, finer details, like code reviews of validation code, may be important. If you're working in a high-security environment (like a bank, defense contractor, certain government agencies, etc.), you should definitely rely on your own employer's security department, security contractors, etc., rather than advice from random strangers on a Web forum.