It's simple enough to explain here.

"Extract points" in this context means dropping the "1." from the gravities and just dealing with the decimal portion as if it were the whole number. I usually call them gravity points, but they mean the same thing.

"I need to add 17 extract points to my 1.063 beer" means he wants to raise his 1.063 wort to 1.080 (80 - 63 = 17).

You measure the total points in your fermenter, so you multiple those extract points (63) by the volume of wort (6). That's the 378.

The equation is set up properly. Adding 1 quart of water raises the volume to 6.25 gallons. So he's trying to find how many total points does he need to add to his existing 378 to give a gravity of 1.080 in 6.25 gallons of beer. Solving for X gives 122.

You can add those points different ways. Dry malt extract has 45 gravity points per pound. So divide 45 into 122 to get the number of pounds of DME to use (2.71). Liquid malt extract is around 36 points per pound.

Note that this assumes the DME takes up no volume, but I'm assuming the liquid will be boiled so the final volume of the portion added is about 1 quart.