README has 6 letters though, so README or README.TXT would have been perfectly OK.

READ.ME just looks cool, same thing as with internet domains.

Edit: I even remember seeing this somewhere as a child and admiring how clever it was (Commander Keen maybe?).

And the capitalization thing in this case is of course also a DOS thing (as opposed to the newer traditions pointed out by another comment.

Tangent: on DOS and Windows, filenames without a file extension suffix were just weird. I mostly expected them to contain binary data :D