You are misunderstanding how open source software development works. Not just anyone can modify the kernel, so no "unknown developers from around the world" are submitting mysterious lines of code. In order to make changes directly, you have to be hired by the Linux Foundation or another organization that has access to commit privileges, not just pass a brief check. Everyone else can only suggest improvements (often in the form of a patch) on the public mailing lists, and hope that the official developers will accept their suggestions and make the changes on their behalf. The review of patches is quite intense and patches often have to go through many revisions before being accepted, sometimes with each line being criticized (often for the most trivial things).
This is not so different from closed source development. You could argue that anyone can make modifications to Microsoft Windows simply by looking for a job at Microsoft. The only difference is that open source software is more transparent about their development process.
With all of this code being submitted by unknown developers throughout the world, how can the security administrators make sure that there isn't any backdoor scripts being added onto the OS?
Everything added is reviewed. The only way to get a backdoor in would be to provide a patch that looks good, but which has an exploitable bug that you are aware of. This is possible, but quite unlikely. Developers don't need to be malicious to introduce bugs at staggering rates!
Aren't there like 15 million lines of code?
More, actually. However a large portion of this is for device drivers which you will never even use. The Linux kernel is monolithic, which means hardware-related code is put in the kernel. The core of the kernel itself is actually not that big. It only looks big when you count the fact that it also includes code from obscure karaoke microphone drivers that are not even present in your kernel.
Do the admins of the OS really understand every line of code and know for sure that people haven't submitted something that could corrupt the system such as malware, backdoors or spyware?
No, although some particularly savvy sysadmins do have a good knowledge of how the kernel works internally. There are quite a few eyes on the code in the Linux kernel looking for bugs. It is hard for them all to hide. Yes, it is possible that a bug was secretly introduced (after all, it was attempted and foiled once), but the painful truth is, such intentional bugs would be a drop in the water compared to all the naturally existing bugs that find their way into such a huge codebase.