How does the booting process with these scripts work?

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

SysVinit (which we'll call init from now on) works using a runlevels scheme. There are 7 (from 0 to 6) runlevels (actually, there are more runlevels 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 runlevel is 3. Here are the descriptions of the different runlevels 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 runlevels is init <runlevel> where <runlevel> is the target runlevel. For example, to reboot the computer, a user would issue the init 6 command. The reboot command is just an alias, as is the halt command an alias to init 0.

The /etc/init.d/rcS script is run at every startup of the computer, before any runlevel is executed and runs the scripts listed in /etc/rcS.d

There are a number of directories under /etc that look like like rc?.d where ? is the number of the runlevel and rcS.d. A user might take a look at one of them (after this chapter is finished, right now there's nothing there yet). There are a number of symbolic links. Some begin with an K, the others begin with an S, and all of them have three 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 000 to 999; the lower the number the sooner it gets executed. When init switches to another runlevel, the appropriate services get killed and others get started.

The real scripts are in /etc/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/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 a S link is encountered, the appropriate script is run with the start argument.

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 LFS system, not ours). The files here are just an example of how it can be done in a nice way (well what we consider nice anyway. You may hate it).