The way he does it, with the angle brackets? That's the easiest way, and by far the weakest. The strongest way is to notch out 2x12's (ideally at least 3, with one in the middle). Oh, and no step can be more than 1/8" difference in rise vs any other stair. This much harder, though.
Combine this with the math to know how many steps you need, and what angle the stairs need to lean overall (this one he touches on), and it's a real pain. Fail, and people get hurt. Your body is pretty sensitive to stairs being perfect.