How does the booting process with these scripts work?

Linux uses a special booting facility named SysVinit. It's based on a concept of run-levels. It can be widely different from one system to another, so it can't be assumed that because things worked in <insert distro name> they should work like that in LFS too. LFS has its own way of doing things, but it respects generally accepted standards.

SysVinit (which we'll call init from now on) works using a run-levels scheme. There are 7 (from 0 to 6) run-levels (actually, there are more run-levels but they are for special cases and generally not used. The init man page describes those details), and each one of those corresponds to the things the computer is supposed to do when it starts up. The default run-level is 3. Here are the descriptions of the different run-levels as they are often implemented:

0: halt the computer
1: single-user mode
2: multi-user mode without networking
3: multi-user mode with networking
4: reserved for customization, otherwise does the same as 3
5: same as 4, it is usually used for GUI login (like X's xdm or KDE's kdm)
6: reboot the computer

The command used to change run-levels is init <runlevel> where <runlevel> is the target run-level. For example, to reboot the computer, a user would issue the init 6 command. The reboot command is just an alias for it, as is the halt command an alias for init 0.

There are a number of directories under /etc/rc.d that look like like rc?.d (where ? is the number of the run-level) and rcsysinit.d all containing a number of symbolic links. Some begin with a K, the others begin with an S, and all of them have two numbers following the initial letter. The K means to stop (kill) a service, and the S means to start a service. The numbers determine the order in which the scripts are run, from 00 to 99; the lower the number the sooner it gets executed. When init switches to another run-level, the appropriate services get killed and others get started.

The real scripts are in /etc/rc.d/init.d. They do all the work, and the symlinks all point to them. Killing links and starting links point to the same script in /etc/rc.d/init.d. That's because the scripts can be called with different parameters like start, stop, restart, reload, status. When a K link is encountered, the appropriate script is run with the stop argument. When an S link is encountered, the appropriate script is run with the start argument.

There is one exception. Links that start with an S in the rc0.d and rc6.d directories will not cause anything to be started. They will be called with the parameter stop to stop something. The logic behind it is that when you are going to reboot or halt the system, you don't want to start anything, only stop the system.

These are descriptions of what the arguments make the scripts do:

Feel free to modify the way the boot process works (after all, it's your own LFS system). The files given here are just an example of how it can be done in a nice way (well, what we consider nice -- you may hate it).