Is there some sort of formula to determine how many bricks I would need to build a completely new structure?
A standard brick is 8" x 2.25" x 3.5".  Ignore the 3.5" as that's how thick the wall will be.  8" * 2.25" is 18 sq. in.  A square foot is 144 sq. in. (12 * 12).  144 sq. in. divided by 18 sq. in. is 8.  But I left out the mortar.  It's possible that the mortar on six bricks is equal to about two bricks.  

Another way of looking at this is that each brick is two thirds of a foot.  So you need a brick and a half to make a foot.  If you stack bricks four high, that's nine inches without mortar.  Add three quarters of an inch of mortar to each brick, and four bricks is exactly a foot.  Now, I think that three quarters of an inch is a bit much, but you also have sideways mortar that I didn't include.  

Anyway, if we assume they know what they're talking about, you have one square foot of wall is six bricks.  If you want your wall to be seven feet high and ten feet wide, you need 7 * 10 * 6 = 420 bricks.  That's $546.  

This answer did the same thing with four 8' tall 14' long walls and got a total of $3494.40.  

I think that the confusion is that you were trying to make a cubic foot wall.  But that's not what they mean.  They mean that a square foot of wall surface takes six bricks.  So when looking at a piece of wall and measuring out a square with one foot to a side, you'll count about six bricks.  Roughly four rows of one and a half bricks each.