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I understand that seven men most be on the line of scrimmage for the offense at the snap of every play. Growing up I've heard this penalty called a multitude of times and recently I began pondering why in the world it is a rule in the first place. I understand that a lot of rules come into place for either two reasons: competitive balance or safety.
What is the purpose of this particular formation rule?
The requirement of seven offensive players on the line of scrimmage was an outcome of trying to make football 'a more open game.' (see The Forward Pass in Football)
To summarize the issue with football in the early twentieth century, the game was low-scoring, and most plays consisted of handing the ball to one player and having a him surrounded by his ten teammates (usually resulting in a gain of less than five yards).
A recent book with a chapter that discusses the evolution of 'a more open game' is Chapter 2 of Newton's Football: The Science Behind America's Game. Coincidentally, the text from this chapter argues that the primary reason that rules were changed was because football was a very dangerous game in the late 1890s and early 1900s (in particular, the Flying Wedge formation).
In addition to the historical reasons, the reason that not only do you have to have 7 men on the line, but you have to have a particular seven men, is to avoid confusion about who is eligible to catch a pass. From the rulebook, rule 7 section 5:
Section 5 Position of Players at the Snap
Article 1: Offensive Team. The offensive team must be in compliance with the following at the snap:
(a) It must have seven or more players on its line (3-19-1); and
(b) Eligible receivers must be on both ends of the line, and all of the players on
the line between them must be ineligible receivers.
(c) No player may be out of bounds.
Thus, not only must seven men be on the line, but those seven must be the 5 ineligible receivers (the offensive line, who may not be the first to touch a pass, and must stay behind the line on passing plays prior to the pass being thrown) with one eligible receiver on each side of them. You'll hear fouls periodically saying "The receiver was covering up the tight end" and similar, which means a WR was on the line and a TE inside of that WR was off the line (typically because the TE went in motion and the two of them didn't properly reset).
That helps the defense, as they then know which five players they don't have to cover in pass defense. Otherwise it would sometimes be confusing as to which of the 6 linemen was eligible without looking at their numbers (which isn't always possible when they're in their stance).
