Why using character ^^ff before every chapter in the TeXBook source

It is definitely not 0xFF, but “Control-L” or ASCII 0x0C (decimal 12), called “form feed” (this is why your viewer shows it as FF).

As you probably know, Knuth started using computers when they weren't like the ones we use today. When the teletype was the only interface, character Control-L was used to tell the teletype “advance to the end of the form”, so the roll of paper could be easily torn up and the session could restart from a fresh state.

Later, screen editors used “Control-L” to tell the computer “do a new page”, so that scrolling past “Control-L” would put the next copy at the top of the screen. It's also useful for marking relevant points in the text, so “search for Control-L” leads, in the case of texbook.tex, to the start of the next chapter.

The Plain format defines

\catcode`\^^L=\active \outer\def^^L{\par} % ascii form-feed is "\outer\par"

so to get yet another check for unbalanced braces. Look at the start of chapter 2:

Technique! The very word is like the shriek
Of outraged Art. It is the idiot name
Given to effort by those who are too weak,
Too weary, or too dull to play the game.
\author LEONARD ^{BACON}, {\sl Sophia Trenton\/} (1920) % composed at Stanford

\eject
^^L\beginchapter Chapter 2. Book Printing\\versus\\Ordinary Typing

Forgetting the year in the citation would make TeX find ^^L during the scan for the argument to \author, but active ^^L is an outer macro.

During the run of TeX on the file it just produces \par, but it could be redefined in other ways.

It's an old control character for a page break (form feed), used long time ago for marking the new page for printers.

It's 0x0C in HEX, also represented as Control + L. WinEdt shows it as ^L - underlined L. Notepad++ show it as FF.