Why is this organization even relevant? Which persons involved with the Linux kernel asked for such a foundation, and what was their justification for it?
The Linux Foundation does several things for the community:
1. Pays Linus Torvalds to work on the Linux Kernel. He initially worked for Transmeta, but then when they let him go he was quickly put on the dole by OSDL (now Linux Foundation) in order to help keep him vendor neutral and allow him to focus solely on the Linux Kernel. (While at Transmeta he had some other responsibilities for Transmeta if I'm not mistaken, so most but not all of his time was on the Linux Kernel.)
2. Helps protect the Linux Trademark that Linus officially owns. Linus did not originally trademark the term "Linux"; then someone did and brought a suite against him, so the community (and corporations) stood up, defended it, and then trademarked it, officially giving Linus the ownership. However, Linus is in now way financially capable of defending it against sufficiently funded groups, so having an organization like Linux Foundation help in that respect is very good.
3. Helps show sponsorship of the Linux Kernel. Companies - especially big companies - like to get tax write-offs. By donating to the Linux Foundation (a charity) they get write-offs and they get to build some good will by having their name publicized as a sponsor.
4. Training - Linux Foundation officially does some training, and support. For example, they help companies get into the Kernel Development process, providing access to key developers, and mentoring on how to get contributions accepted. Greg Kroah-Hartman has been quite helpful to a number of companies in that respect; that doesn't mean they get a straight line into having their patches accepted, but that they get mentored on what to do so the patches are *likely* to be accepted - thus more hardware and features are supported by the Linux Kernel.
There's more they do as well, but those are the biggies.